Say Hulagu doesn't return to Mongolia in 1260 (can be Mongke Khan lives longer, or Ariq Böke doesn't try to seize power, or whatever works) -- and that he stays in the West long enough to defeat the Marmelukes and, if not technically conquer and rule, then at the least neutralize Egypt (AAR, the Nile runs red with blood and black with ink, and all the rest).
My question -- what does Hulagu do next? Does he stick around to try to subjugate Egypt thoroughly? Does he move east to consolidate his rule in the Ilkhanate?
Or do his conquests continue, and if so, where? Does he move against the Crusader States? Does he seek out grassier areas for his armies to move into, like Antalonia? If so, is it possible he tries to march on Constantinople? Or -- and this sounds pretty crazy to me, but I've seen it suggested here before -- do his armies move south or west, into Africa?