I think the argument could go both ways.
The obvious choice is to shove all-in because we are fairly short stacked and a pocket pair is solid if we are heads up.
However, if we are called, we are usually just on the better side of the coin flip - yet still
gambling. With this in mind, you may want to fold the pocket 4 hand if you believe you are the stronger player with chances to get it in at a better spot (although there is no guarantee the cards will cooperate with the blinds coming for you soon).
What I would avoid is the tempting idea of calling and shoving or folding based on what they do because we lose fold equity from not shoving right away and most good players will put you all-in anyhow (hence forcing you off the hand even if they had nothing).
I guess there are several factors I'd consider apart from the obvious in the specifics of this hand.
What things I'd consider may include:
-Our perceived skill compared to our competition with us more willing to fold if we are better players
-This sounds like a regular MTT, but if the format is turbo or faster, then the blinds increasing may get us to shove more often here so we avoid being blinded out
-Similar to the last point, are antes in play yet? Everyone is answering this question without antes I think, but with antes in play, then we can always be content with shoving 44 at 15bb.
This is a good question, but I would also be curious in asking: "How did you get to 15 bb in the first place?" It could be a number of reasons including a bad run of cards, a bad beat/cooler against us, playing too loose or many other things: but this question is always worth investigating because sometimes the real "mistakes" can be found there.