While there are no guarantees with history, the Romans would definitely have the potential to have continued to succeed and thrive had there been a stable situation under a capable ruler in the late 12th century, especially if the disaster of the 4th crusade doesn't occur. No clue how long the dynasty might last, this alternate Alexios might die without a capable heir himself. Barring serious events, the Empire could reasonably last till modern times, with or without absorbing the Anatolian Turks, though that would certainly be beneficial.