Late in 582 Emperor Tiberius II became gravely ill. He appointed Maurice and Germanus as his heirs and each was engaged to one of Tiberius' daughters and elevated to the rank of Caesar. Historians feel that Tiberius initially intended to divide the empire into two, with Germanus controlling the West while Maurice controlled the East. On the 13th of August, however, Tiberius crowned only Maurice as Augustus. The next day Tiberius died and Maurice became sole emperor...
WI Tiberius II had indeed divided the Empire in 582???
I suspect it wouldn't have lasted for very long. There really wasn't all that much "west" under Roman/Byzantine control by this point, and, with the exception of Carthage (which would probably wind up being the seat of the western imperial court) and its environs, none of it was economically worthwhile. The best case scenario is that the situation persists only until the western provinces start falling at the start of the seventh century, and the worst case is civil war between the two emperors.
Regardless of who wins this hypothetical war, a unified imperial court is likely to return to Constantinople. There are all sorts of nasty side-effects the fight might have, as well. Italy, already badly scarred from Justinian's reconquest, is going to fare very poorly if it gets involved, for example. And the entire empire is likely to be in much worse shape for its showdown against Persia in the coming years. I can quite easily see the empire falling apart in the next round of wars against Persia. From there, there's all sorts of interesting things that could happen: if Persia isn't as badly damaged by Heraclius's counterstrike, than it probably holds up much better against the initial Muslim conquests. If Islam isn't butterflied away entirely, it probably takes a very different form, perhaps moving west initially and picking off a much weakened Byzantine remnant while Persia survives and maintains its Zoroastrian faith. A much weakened, ever-more regionalized Roman state might persist in parts of Greece and Italy, but it's not going to be a major power anytime soon.