"There would be even more possibilities if promoted pieces had the same value as the original ones. In this case promoting in general would become less compelling, and underpromoting could become the norm."
That would actually limit possibilities in high-level play, since no player would want to advance their pawns for promotion for the simple reason of keeping potential points away from the opponent. I would rather give up 1 point than give up 3, 5, or 9.
In regular chess underpromoting a pawn to rook or bishop is rarely beneficial, and only serves as a tactical mean to avoid stalemating your opponent. Not so in 4-player chess. The range of possibilities is much wider. You could opt to underpromote for at least the following reasons:
1) To avoid threatening an opponent. If your new queen is attacking an opponent's queen, and he has another piece under attack, he'll probably prefer to save his queen by capturing your new queen. But if you promote to a non-threatening piece, your piece may survive!
2) To avoid recapture by a pawn. If your new queen can be captured by a pawn, she WILL be captured more often than not. But an advanced pawn is valuable by itself, so your opponent may decide not to exchange it with a lesser piece.
3) To avoid checking an opponent. A non-threatening opponent may be ready to wreck havoc at the base of the top dog. You don't want to disturb this "friend" by checking him. So you can underpromote to a non-checking piece.
There would be even more possibilities if promoted pieces had the same value as the original ones. In this case promoting in general would become less compelling, and underpromoting could become the norm.