Nah I really don't think 44 makes almost any sense here. So he's bluffing
the whole way here right? I mean he's got to figure that very little that he beats is calling flop (AK, but you have tons of equity). So, given that he has to be bluffing, we can evaluate his line.
His cbet is OK given that he can maybe fold out something like 55 or 66. After this, the K turn smacks your floating range, right? If he's betting turn, he has to bet river to get any other PPs that were hanging around to fold. So pretty much he has to either A. Check-fold flop. B. C-bet-give up. C. Triple barrel. Betting two streets is super meh. And triple barreling is meh unless you're super tight and would always fold pairs to 3 bets.
tl;dr. Don't take his line with 4s.