Fold My Set?

T10000 to start
Assume villain to be a very good player

Hero T12700 (CO)
Villain T8300 (BB)
50/100 Blinds

Hero dealt 4h4s

Two limpers, hero limps, villain raises to 400, limpers fold, hero calls.

Flop 3h,4c,5h

Villain bets 800, hero raises to 2000, villain calls

Turn Ad

Villain is allin, hero????

My thinking was that he had AK,AQ, or 88-JJ
In retrospect, I'm not so sure my play on the flop was very good.

Would he go all in with TP or 2 pair?
Any chance he played AA this way?

Comments appreciated.