A very similar game to what you propose (or possibly the exact same, depending on how you’re ‘overlapping’) was(is?) sold as Chess[sup]4[/sup]. The area where each player starts is four deep, and then the middle area is a regular sized board.
The rules of that game where that pawn promotion could be on any terminal edge (that is, at the back of the three starting areas for the other players. Theoretically a pawn could be stuck (along the right player’s right sunken edge, or the left player’s left sunken edge; perhaps promotion was allowed then). In any case, promotion was exceedingly rare.
Some general observations :
The game as a whole becomes more tactical, especially in an ‘every man for himself’ game.
Bishops are very powerful; at least as powerful as rooks. In the four-deep variation, they start out in a vastly stronger position.
Knights are correspondingly weakened, but make good defensive players, and their cornering ability is a plus.
The center is nearly worthless, unless it comes down to an endgame, and that is rare.
Four-player chess is amenable to team play (and in my opinion the best way to play):
Play proceeds around the board in order, alternating teammates (clockwise is traditional).
In team play, your partner sits opposite you, and you cannot speak to them during the game (at least as far as collaboration is concerned; we usually allowed oblique chatter as long as it was not your team’s move).
Checkmate (on you) does not occur until the player just before you has moved. This means that your partner can ‘save’ you from checkmate from the player after you (but they still must move out of check if in it, and you can’t deliberately enter check even if your teammate could save you).
The first player on a team to lose their king is not eliminated. Instead, they use a move to remove the king, and then continue to play as normal, only losing if their teammate loses. This is the rule that makes the game interesting.
Another four-player version (tougher to play, but a bit more interesting) is Doubles Chess. Imagine a board with four cuts from the center to the side dividing it into quadrants. A similarly cut board is spliced to it (imagine them as intersecting at right angles, and that you can move ‘diagonally’ to the opposite side of ‘your board’, and that’s sort of it). This was sold as a 2-d board with the squares distorted, and it’s really hard to figure out how to properly move.