I've been developing a theory in which King Bela III of Hungary became Roman Emperor. Béla was the second son of King Géza II of Hungary. In 1164, Emperor Manuel I of the Romans signed a treaty with the new King of Hungary, Stephen III, and Bela was sent to be educated in the Byzantine court. Emperor Manuel saw a true potential in Bela and, as he had no sons, married him to his daughter, Maria Komnena, thus making him heir to the throne. Manuel created the title of despotes, seconded only by the Emperor, and granted it to Bela. Bela was also given a Byzantine name, Alexius (according to the AIMA prophecy). However, in 1166 Manuel finally got a son called Alexius and he cancelled the engagment of Bela-Alexius and Maria, depriving him of his title of despotes. In 1172, Bela's brother, King Stephen III of Hungary, died childless and Béla returned to Hungary to claim the throne. He later died in 1196, aged 47/48, being one of the best Kings of Hungary. Ever.
So, let's say that ITTL (in this time-line, as we call it on alternatehistory.com) Alexius II Komnenos was never born and that King Stephen III lived longer (quite possible. He died with 24 years). That way Manuel I dies without a male son and Béla becomes Emperor Alexius II of the Romans. That means no Andronikos Komnenos as Emperor and no Isaac II/Alexius III. There will be no Latin Massacre of Constantinople and the 4th Crusade could be butterflied away. What would happen?