Interesting. With Maurice still alive what can we expect when the Arabs come along?
Well, Maurice will be dead by time the Arabs are on the scene, so the Emperor will probably be his son Theodosius, who was born in 583. We don't know a lot about Theodosius, so how his reign would pan out is unclear. Here, though, are a few thoughts about how a Roman Empire in which Maurice lives until about 605, will look at the time of Mohammed's death.
- The Empire may very well be split into Eastern and Western portions, given Maurice seems to have intended to make his second son Tiberius Emperor of the West, as well as giving his other sons some small parts of land. I suspect, however, due to the relative youth of the sons at the time, that only Theodosius and Tiberius will be able to take effective control. A civil war at some stage wouldn't surprise me, in which Theodosius will almost certainly come out on top of.
- There'll probably be another Persian war at some stage, though the Empire is unlikely to collapse as easily as it did IOTL, due to there being a strong and legitimate dynasty on the throne. The parsimony of Maurice's years may have eased a bit by, say, the mid 610s if this is when the war breaks out, assuming Maurice spends the last few years of his reign thoroughly beating back the Slavs and Avars. In addition to this, a mini-economic boom will probably happen in the first decade of the 600s, as the economy recovers from the plague of 591. The Persians certainly shouldn't be underestimated, but they're unlikely to find an invasion such a walkover as OTL.
- Psychologically, the Empire will be a different place without the devastation of the Persian war, and the trauma of having Persian armies encamped outside Constantinople. Byzantine Christianity will probably take a slightly different path from OTL, and the Emperors won't be quite so fatalistic- I think there's a lot to be said for the notion that it was the huge shock of such a complete collapse that prompted the shift from Justinian-style classical Roman glory to the rather arcane squabblings and more limited world view of the later Emperors.
- At some stage or another, religious problems will have to be dealt with. Maurice's general policy was, IIRC, to be generally tolerant, whilst coming down like a tonne of bricks on troublemakers. Whether this will really be tenable for Theodosius is debatable, given the dependence of the Roman economy on Monophysite Egypt. I would guess that a compromise doctrine will have to be worked out at some stage or another, and to do so, the Papacy will have to be antagonised. Which could mean a Papal rebellion against the Emperor...
All in all, Theodosius III will have a hell of a lot in his in tray to deal with, but his prospects are not completely bleak- he's probably inheriting a better situation for the ERE for anyone since Justinian. The future could well be very bright for him, if he sticks to the course of relatively decent rule.