Pawns, pawns, and more pawns. It's very tricky to alter the pawn because it keeps the balance of chess intact, but that hasn't stopped folk from trying. I came up with all different types of pawns for a 'choose your own army' variant I made a while ago. To name a few:
Crab pawn - Moves as a regular pawn, but instead of moving 1 square diagonally forwards to capture, it makes a narrow knight move instead. For example, if a crab pawn is on e4, it controls g5 and c5. Weaker than the regular pawn because it can't attack anything adjacent to it. Similar to your knight pawn, but the crab pawn moves like a regular old pawn.
Wizard's pawn - Moves as a regular pawn but no initial double-step move. Captures 1 or 2 squares diagonally forward, then immediately returns to the square it attacked from. A much stronger pawn, for sure. Takes 2 moves to get to the centre but will hold its ground quite nicely. A handy piece for cracking open closed positions.
Blitzkrieg pawn - Moves and captures like a shogi pawn, but upon capturing, immediately promotes to the piece it just captured (unless it's a pawn-type piece in which case it remains a blitzkrieg pawn). A quick promoter but doesn't possess a lot of power.
Chess 2 features a new kind of pawn for one of its armies, the nemesis pawn. It has no initial double-step move but it can, at any time, move 1 square towards the enemy king without capturing. Since midline invasion exists in Chess 2, it pairs up well with the eponymous nemesis (a non-capturing queen which can't be captured but can still give check and mate) and is good at harassing the enemy king, keeping it away from the middle of the board.
Then you have these musketeer chess 'superpawns' which have been floating around for a while. They all appear to be different, some have additional double-step options (I believe one can leap like a shogi knight?) but what they all have in common is that once they cross the middle of the board, they gain new movement properties. They're reminiscent of xiangqi pawns in that both 'promote' upon reaching a certain rank on the board.
But changing the pawn's moves isn't the only thing you can do with them - you can also change their ability to capture en passant. En passant is pretty situational as it is, and feels a bit inelegant. The pawn on the other hand, while it can be changed, is fine as it is. @game_designer proposes a really great fix to en passant that not only simplifies the rule but also makes it more apparent for newer players - a pawn cannot move 2 squares on its first move if it would end up adjacent to an enemy pawn. For example, if White's pawn is on e5, Black cannot play d5 or f5 because White's pawn controls d6 and f6. In doing this, en passant no longer exists but instead a rule which is much more understandable takes its place.
This is basically a list of my invented pieces, specifically, of the pawn type.
Squire:
Squire is basically a wazir (moves and captures moving one step ortogonally) able to teleport to the adjacent squares of any high piece (of no pawn type).
Phalange pawn:
Phalange pawn are basically common pawns, but with an special ability.
When several phalange pawns (two or more) are placed next to each other (aligned and horizontally), the entire group of phalange pawns can move up to two steps per turn at the same time.
Barbarian:
Barbarian are ferz (moves and captures moving one diagonal step) able to capture until three pieces at the same turn (that mean, three capture movements (doesn't have why be aligned) at the same turn).
Archer:
Pawn able to capture without move, as well as capture moving.
Knight pawn (minor knight):
It is pawn which movement is a leap of two steps forward and one step horizontally; and its capture is a leap of two steps horizontally and one step forward.
Hive mind pawn:
Hive mind pawn can move individually, or you can move several or all the hive mind pawns at the same turn.