How would chess change if you could self capture?

It would make the game more tactical. For example if white castles early then ...Rxh7 would immediately activate the rook as an attacker. For the same reason minor pieces would never be inactive for very long.

So overall a bad rule change. A slight increase in tactics while detracting massively from strategy and positional play.

I love thinking of all the different variations we could have. How about a game with no pawns? Sometimes they just seem to be in the way. All Queens and a King? Option to move your own or one of your opponents pieces? I invented Battlechess with 4 boards and 3 armies on each side.

1.Kxe3.

Capturing your own pieces is a variant known by various names – originally called "Free-Capture chess" when invented in 1971. On Chess.com it's a playable game named Capture Anything Chess. See https://www.chess.com/terms/capture-anything-chess

You already know what white's first move's going to be... 1. Kxd1

It's interesting to think about whether or not it would make the game more complicated.  It definitely wouldn't be as fun though.