Cool!
Suggestion: 4 Player Chess for mobile devices should be like this

This would be a totally different variant, and would have to be rated differently. And PC users wouldn't want to be excluded.

One problem with your starting position: the bishops are all pointing at unprotected pawns. Switch the bishops with the knights.

There are 2 different setups, that is why I posted 2 pictures. The kings are in the corners on one of the boards, but they are near the middle on the 2nd board. We can change it any way we like. It is just some illustrative examples. We can put the queens to the board and make it 9x9, for example.

@ MGleason there is the knight on b1/h2/g8/a7
Um... oops, good catch. Still, that knight can't move until the pawn is either moved or protected.

I'm somewhat inclined to think the original Chaturanga setup is better. The king is really the only piece with the flexibility of movement to counter an early pawn rush from the side. However, the bishops being unprotected and directly across from each other gives an advantage to the first to players, so maybe the bishops shouldn't be in the corners.
The board is huge now 14x14. We need something smaller for mobile devices. So, I propose to make a separate variant of the 4 Player Chess. Hope chess.com developers will add it to their 2018 or 2019 To Do List.
To make the board smaller we can remove some of the pieces. If each player has 1 king, 1 queen, 1 rook, 1 bishop, 1 knight + 5 pawns the "cross" board can be 11x11 or 9x9.
Another way is to make a square board and something like this:
This picture is a setup of the ancient game called Chaturanga, the mother of Chess (if you did not know).