Here's one where I raise a strong QQ and am up against AA. Re-raising wouldn't help him either, because I view my QQ as strong enough to call and I have position. This player was a marginal winner in my db before this hand (I have 667 hands against him), but this one put him as marginal loser. He's pretty tight, but tends to overplay hands.

Since the flop was seen HU, I might have made a small bet on my QQ as overpair, but not pot. As it plays, I view myself here as protecting my trips with the full pot bet. My current view, at least, is that full pot will mean a medium-strong hand like this one. Lower bets either mean a very weak hand (such as mere overpair on an relatively uncoordinated board in LP) or else an extremely strong one (such as top set with flush redraw).

My opponent obviously views my bet as simple CB and tries to check-raise me off of it.

PokerStars Pot-Limit Omaha High, $2 BB (9 handed)

MP1 ($78)
MP2 ($79.55)
Hero ($496.90)
CO ($195)
Button ($196)
SB ($185.10)
BB ($321.20)
UTG ($232.70)
UTG+1 ($216.65)

Preflop: Hero is MP3 with , , , .
1 fold, UTG+1 calls $2, 2 folds, Hero raises to $9, 4 folds, UTG+1 calls $7.

Flop: ($21) , , (2 players)
UTG+1 checks, Hero bets $19.95, UTG+1 raises to $54, Hero calls $34.05.

Turn: ($129) (2 players)
UTG+1 checks, Hero checks.

River: ($129) (2 players)
UTG+1 checks, Hero checks.

Final Pot: $129

Results in white below:
UTG+1 doesn't show.

Outcome: Hero wins $129.

UTG+1 has AsAd2d3c

I don't really think he has enough hand to check-raise. I think he should bet out and fold to my raise.

yeah but if he leads out, you could see if he's really serious by calling and seeing what he does on the turn. By check raising, he's trying to show you that he's really serious about the hand and putting even more pressure on you to fold. You would have to have a 9 with a good kicker to continue after his check raise while you don't have to if he leads out and then he has to be oop if you call the lead.

Yeah, i don't think he's check-raising for value, I think he's just making a play, assuming you'll CB with anything and will fold anything that doesn't have a 9 or TT in it. Against the right opponent I reckon it can be mildly +EV.

It's not something I tend to do (especially as, because he has two aces, your raising hand is quite likely to have made something on that flop) and usually just lead and fold to a raise as you said, but against the right mark I guess it can work. If you're the sort liable to call this flop with a straight/flush draw but fold the same to a CR I guess that could be another feather in its cap. Not how I'd play it but I think you'll see AA played much worse.