Seems to me this could only work with single-player games (which appears to be the case in the sample video). And lag in single-player is only an issue thanks to Blizzard for forcing people to play on Battle.Net, even during single-player campaigns. So thanks, Blizzard, for fixing an issue you introduced with your over-zealous efforts to curb piracy.
With multi-player, actions by each player must be arbitrated somehow, and that's usually done by the server. Otherwise, you might have one person killing a monster at the same time another person is getting killed (or in PVP, two players killing each other at the same time, how would that work?). Imagine if you make the killing blow to a monster on your computer, but on your friends computer, the monster casts some spell that kills you instead, how would this be resolved on the server? For that matter, how does each client end up fighting the same monster? If each client computer makes it's own decision, wouldn't you technically be fighting two different monsters, a copy for each person in the party? I don't think that could work, so I don't think this solves lag where it's traditionally most annoying, multi-player (traditionally, because traditionally, single-player don't got lag!!!).