Would certain checkmates be possible if the king could move into check? Similar to a previous question of mine, let's suppose there's a piece that moves like the king, but isn't restricted from moving into check.

While a normal king and two knights can't force checkmate, can this "king" and two knights force it by themselves? What about a "king" and a single knight or bishop?
You could force a checkmate with two knights and a "king". Because the "king" does not fear being taken, it can move directly in front of the opponent's king, if it is protected by a knight.  All it has to do is do this when the opponent's king is on the side of the board and thus cannot retreat - and it is known that two knights and a normal king can force the opponent's king to the side of the board.

Based on trying it out, I believe a "king" and single knight could also force checkmate. The knight supports the "king" very well, cutting off squares and proving good places for the "king" to advance. And when the knight retreats, the "king" is usually ready to go into that same square if necessary, cutting off the king and forcing it backwards.

With a bishop and a "king", I think the question is whether the king can be forced into a corner which is the same color as the bishop. The following position, with either side to move, results in a quick checkmate, because Black must go into the corner:

[FEN "1k6/8/1K2B3/8/8/8/8/8 b - - 0 1"]


I believe this can't be done from many positions, though.  From most positions, the king escapes into the wrong corner. Although it can be driven from that corner, it can then flee to the wrong corner on the opposite side of the board.