I have read the suttas but can’t find a clear answer. 
SN 44.6:
On one occasion Ven. Sariputta and Ven. Maha Kotthita were staying near Varanasi in the Game Refuge at Isipatana. Then in the evening, Ven. Sariputta emerged from his seclusion and went to Ven. Maha Kotthita and exchanged courteous greetings with him. After an exchange of friendly greetings & courtesies, he sat to one side. As he was sitting there, he said to Ven. Maha Kotthita, “Now then, friend Kotthita, does the Tathagata exist after death?”
“That, friend, has not been declared by the Blessed One: ‘The Tathagata exists after death.’”
“Well then, friend Kotthita, does the Tathagata not exist after death?”
“Friend, that too has not been declared by the Blessed One: ‘The Tathagata does not exist after death.’”
“Then does the Tathagata both exist and not exist after death?”
“That has not been declared by the Blessed One: ‘The Tathagata both exists and does not exist after death.’”
“Well then, does the Tathagata neither exist nor not exist after death?”
“That too has not been declared by the Blessed One: ‘The Tathagata neither exists nor does not exist after death.’”
“Now, friend Kotthita, when asked if the Tathagata exists after death, you say, ‘That has not been declared by the Blessed One: “The Tathagata exists after death.”’ When asked if the Tathagata does not exist after death… both exists and does not exist after death… neither exists nor does not exist after death, you say, ‘That too has not been declared by the Blessed One: “The Tathagata neither exists nor does not exist after death.”’ Now, what is the cause, what is the reason, why that has not been declared by the Blessed One?”
(i. The aggregates)
"For one who loves form, who is fond of form, who cherishes form, who does not know or see, as it actually is present, the cessation of form, there occurs the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death.’"For one who loves feeling…
"For one who loves perception…
"For one who loves fabrications…
"For one who loves consciousness, who is fond of consciousness, who cherishes consciousness, who does not know or see, as it actually is present, the cessation of consciousness, there occurs the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death.’
"But for one who doesn’t love form, who isn’t fond of form, who doesn’t cherish form, who knows & sees, as it actually is present, the cessation of form, the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death’ doesn’t occur.
"For one who doesn’t love feeling…
"For one who doesn’t love perception…
"For one who doesn’t love fabrication…
"For one who doesn’t love consciousness, who isn’t fond of consciousness, who doesn’t cherish consciousness, who knows & sees, as it actually is present, the cessation of consciousness, the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death’ doesn’t occur.
“This is the cause, this is the reason, why that has not been declared by the Blessed One.”
(ii. Becoming)
“But, my friend, would there another line of reasoning, in line with which that has not been declared by the Blessed One?”"There would, my friend. "For one who loves becoming, who is fond of becoming, who cherishes becoming, who does not know or see, as it actually is present, the cessation of becoming, there occurs the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death.’
"But for one who doesn’t love becoming, who isn’t fond of becoming, who doesn’t cherish becoming, who knows & sees, as it actually is present, the cessation of becoming, the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death’ doesn’t occur.
“This, too, is a line of reasoning in line with which that has not been declared by the Blessed One.”
(iii. Clinging/sustenance)
“But, my friend, would there another line of reasoning, in line with which that has not been declared by the Blessed One?”"There would, my friend. "For one who loves clinging/sustenance, who is fond of clinging/sustenance, who cherishes clinging/sustenance, who does not know or see, as it actually is present, the cessation of clinging/sustenance, there occurs the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death.’
"But for one who doesn’t love clinging/sustenance, who isn’t fond of clinging/sustenance, who doesn’t cherish clinging/sustenance, who knows & sees, as it actually is present, the cessation of clinging/sustenance, the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death’ doesn’t occur.
“This, too, is a line of reasoning in line with which that has not been declared by the Blessed One.”
(iv. Craving)
“But, my friend, would there another line of reasoning, in line with which that has not been declared by the Blessed One?”"There would, my friend. "For one who loves craving, who is fond of craving, who cherishes craving, who does not know or see, as it actually is present, the cessation of craving, there occurs the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death.’
"But for one who doesn’t love craving, who isn’t fond of craving, who doesn’t cherish craving, who knows & sees, as it actually is present, the cessation of craving, the thought, ‘The Tathagata exists after death’ or ‘The Tathagata does not exist after death’ or ‘The Tathagata both exists and does not exist after death’ or ‘The Tathagata neither exists nor does not exist after death’ doesn’t occur.
“This, too, is a line of reasoning in line with which that has not been declared by the Blessed One.”
“But, my friend, would there another line of reasoning, in line with which that has not been declared by the Blessed One?”
“Now, what more do you want, friend Kotthita? When a monk has been freed from the classification of craving, there exists no cycle for describing him.”
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What happens to the Buddha after he dies? The same things as the quotient of dividing by zero.
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This is the main example of the kind of question the Buddha discouraged asking in the Suttas. --Whether the Tathagatha exists, does not exists, or neither does nor does not exist after death.
The Buddha very much tended to avoid answering ontological questions like this; the principal goal running through the suttas is the practical concern with ending suffering.
However, there does seem to be something fundamentally different about the arahant. For everyone else at death (unless one attains arhantship on one’s death bed), there’s rebirth. Maybe not in some fundamental sense (people argue about this), but at least in practical terms the person ends up somewhere else. The Buddha though his divine eye in the suttas, when asked, could tell where such-and-such a person ended up (perhaps in some named heaven realm or other).
The arahant is different (certainly at death). There’s the memorable image from SN 22 of Vakkali who had “used the knife”:
The Blessed One saw in the distance the Venerable Vakkali lying on the bed with his shoulder turned.
Now on that occasion a cloud of smoke, a swirl of darkness, was moving to the east, then to the west, to the north, to the south, upwards, downwards, and to the intermediate quarters. The Blessed One then addressed the bhikkhus thus: “Do you see, bhikkhus, that cloud of smoke, that swirl of darkness, moving to the east, then to the west, to the north, to the south, upwards, downwards, and to the intermediate quarters?”
“Yes, venerable sir.”
“That, bhikkhus, is Mara the Evil One searching for the consciousness of the clansman Vakkali, wondering: ‘Where now has the consciousness of the clansman Vakkali been established?’ However, bhikkhus, with consciousness unestablished, the clansman Vakkali has attained final Nibbāna.”
However, this type of statement does not just to apply to arahants at death. MN 2 talks about even a living arahant being in a sense “untraceable”:
“Bhikkhus, when the gods with Indra, with Brahmā and with Pajāpati seek a bhikkhu who is thus liberated in mind, they do not find anything of which they could say: ‘The consciousness of one thus gone is supported by this.’ Why is that? One thus gone, I say, is untraceable here and now.
Something has gone out or been extinguished in the arahant. Whatever this is isn’t tied up with existence or non-existence (given the “untraceable” arahant is still breathing). And in the very next paragraph of that sutta, the Buddha emphasizes his teaching is not about the extermination/extinction of an existing being:
“So saying, bhikkhus, so proclaiming, I have been baselessly, vainly, falsely, and wrongly misrepresented by some recluses and brahmins thus: ‘The recluse Gotama is one who leads astray; he teaches the annihilation, the destruction, the extermination of an existing being.’ As I am not, as I do not proclaim, so have I been baselessly, vainly, falsely, and wrongly misrepresented by some recluses and brahmins thus: ‘The recluse Gotama is one who leads astray; he teaches the annihilation, the destruction, the extermination of an existing being.’
Now, of course, people do argue that the phrase “an existing being” is a kind of get-out clause here, that perhaps the Buddha was then no longer an “existing being”, so extermination could still apply. However, I tend to think though that something else is being pointed at here. Am not exactly sure what that is. But IMO all four logical possibilities enumerated in AN4.38 somehow no longer apply (the so-called fourfold negation):
Here, whatever ordinary personal truths may be held by ordinary ascetics and brahmins— … ‘The Tathāgata exists after death,’ or ‘The Tathāgata does not exist after death,’ or ‘The Tathāgata both exists and does not exist after death,’ or ‘The Tathāgata neither exists nor does not exist after death’—a bhikkhu has discarded and dispelled them all, given them up, rejected them, let go of them, abandoned and relinquished them.
What’s left after eliminating these four possibilities? I’m not sure! That’s not much of an answer is it! 
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Say that what happens is: nothing else at all, ever. It’s an earned annihilation of experience.
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I suppose I must love becoming then.
But it still seems like a reasonable question to ask, and I still don’t understand why the Buddha didn’t give a straightforward answer to it.
That could be the case, but if so, why didn’t the Buddha just say that?
It’s addressed in SN 44.6, above:
Now, what is the cause, what is the reason, why that has not been declared by the Blessed One?
There’s also a quote by Norman in a Piya Tan article:
We are accustomed to take the word tathāgata as being synonymous with Buddha, but it is very unlikely that all those who were discussing these questions in the passages recorded in the Udāna were actually discussing whether the Buddha lives or does not live after death.
It is noteworthy that the commentarial tradition followed by Dhammapāla explains tathāgata as attā: “Does the self exist after death?” Although the list of questions which we have is clearly a stereotyped one, we can assume that it was based upon questions which occupied the mind of the religious teachers who were contemporary with the Buddha.
The four questions concerning the existence of a tathāgata after death are in fact included among those to which Sañjaya Belatthiputta is reported by Ajātasattu to have given a prevaricating answer.
Somewhat complex.