That got me thinking though. What if instead it is Diablo that Azmodan is talking to, and Leah is seeing it from his perspective. Obviously for one, there has been a lot of speculation going on that Diablo has possessed Leah in the continued story. Therefore, it wouldn't be unrealistic for her to be having visions of him or from his perspective beforehand.
1. Azmodan says "Our brethren", which obviously could be Azmodan simply referring to himself and his brethren. But maaaybee, he was talking to Diablo, saying "our" .. as if Diablo (or him and the prime evils) intentionally tricked the others, hmmm? And thought he was so clever? lol. Otherwise, would he have not simply said "my" brethren? Not to mention, it does turn out of course that the Prime Evils purposely had themselves banished to gain the power of the soulstones. So at the very least, they did trick their other brethren, if not that Diablo himself had some other sinister plans.
2. He says "I defy you", which works for Tyrael, but also could mean he defies Diablo! That makes sense.. they like to do that. Hence the banishing of the Prime Evils and whatnot. (Besides, is there really reason for Azmodan to say "I defy you!" in such a way to Tyrael? That should already be a given.)
3. His minions will pour forth searching for the black soulstone. Maybe he found out the power of the black soulstone that Diablo had his hands on originally and allowed him to survive?
Consider this. Perhaps Diablo sealed part of his soul or something into the "black soulstone", and in order to resurrect himself he intends on someone sticking it in their head or something, simillar to how he corrupted the D1 hero? Now, the video implies that Azmodan intends to find the black soulstone and use its power, but he never specifically states that. He simply says "I know it is the key" (to Diablo returning himself?) and "It will be mine" (to destroy?). It's very possible he simply intends to have his armies ravage the land, find the soulstone, and destroy it to prevent Diablo's return in order for him to be "THE prime evil" instead of Diablo.
After all, what is currently really preventing Azmodan from being THE Prime Evil? He clearly has the majority of hell's power and a nearly endless army already. But if he knows of Diablo's possible return, that would obviously be a hindrance to him, and Diablo would be THE Prime Evil. So in summary, Azmodan defies Diablo, knows he has hidden the black soulstone away with part of him sealed inside it (with plans to return), and intends to ravage the lands and destroy the stone before he can do so.
4. More evidence to support this theory - If you pay attention in the video, Leah has what is presumably the black soulstone directly in front of her as she stares into it and has her vision of Azmodan talking. So clearly if Diablo has sealed himself away into it, plans to possess someone, people think Leah is the one possessed, she's having visions in Diablo's place while looking into it, etc. etc.... it all makes sense!
(Also, notice that before she stares into it and has her visions, as she's flipping through the pages it lingers for a good moment on a picture of Diablo's head, then flips to the stone on the next page. Coincidence?)
Now, I'm not saying this is for sure how it is, and clearly Tyrael is a likely candidate as well for who Azmodan is talking to. But I do think, given the information, it's just as (if not much more) likely that he's speaking to Diablo.