This character is the only character that a week later bothers me. He is completely extraneous. I get the idea behind "why" he needs to exist, but I think there'd be an even more interesting story if he doesn't, because it makes things inherently chaotic/unstable.
As for his powers and motivations, who the fuck knows, man. I think the idea is that motivation wise, he wants to destroy the Analyst who also chained him up like he chained Neo. Why he doesn't kill Neo first is beyond me. Maybe the bullettime ends if Neo is dead and that's the advantage he needed to get the Analyst? As for why he's not subject to bullettime, who the fuck knows. At first I thought, "maybe it's an effect that is centered on Neo and makes everything relative to him slow" but Trinity is also slowed by it. So who the fuck knows. Character shouldn't have been in the movie, imo, and deal with the ramifications of the "unbalanced equation."