We have had variants of this. For example, to be able to promote to a player of the other side in to continue a more even game (great when you play with kids). Or my favorite, in school we could promote a pawn to Obelix - a special piece that could wipe all the other players off the board and onto the floor. Very useful when you have limited time before classes start!
Promote a pawn to a pawn
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Me personally, I would make it so that you can promote into a backwards pawn and when you reach the 1st rank you can become a SUPERQUEEN
There is a situation where you would want to promote to a pawn, which is to make a stalemate trap.
I agree. It's not hard to imagine positions where leaving the pawn there would be an advantage. I think if it were left a pawn though, it wouldn't be a promotion. They would have to call it something else.
Since a pawn reaching a dead end doesn't seem to conflict with the purpose of the game I don't see why it couldn't just stay there. No different than a pawn that's blocked by an opposing pawn.
Maybe the rule could be modified so it's more like en passant. Each time that opportunity happens the player has the option of using it or not. (assuming it doesn't create an illegal position like exposing the king to check). I don't see why a pawn HAS to be promoted. Just like I don't see why a pawn HAS to capture en passant.