In the world of generalized chess, anything goes. Chess variants have always introduced new pieces wherever applicable, and if they befit the game's needs. Not every piece will be as one-track as the pieces I've shown earlier; there's no restriction against going crazy with your pieces.
We will deal with hybrids, pieces made from combining movements from leapers, riders and hoppers. To clarify, if a piece does one leap forward and another kind of leap back, it is simply a compound leaper, for a piece to be a hybrid it has to be able to move in two different movement types.
The Simple Mix
Take one movement type and a different movement type, put them together, and you have a simple hybrid. The most notable examples are the knight hybrids commonly explored in more well-known chess variants. The common names for these are the Chancellor (Rook + Knight), Archbishop (Rook + Bishop) and the Amazon (Knight + Queen). As a piece with a Knight and Queen move can be a real monster, some games either try to neuter the piece or place it on a larger board, but there are also games that let the anarchy of an Amazon dictate play, fun but horribly unbalanced.
Not much can be said here that I haven't spitballed on the previous essays; hybrids function as two pieces at once though no one splits hairs at this level.
One Way, then Another
In orthodox chess a piece moves the same way wherever it is on the board whether it is capturing or not, the only piece not to do this is the pawn.
Pieces whose moves depend on where it is on the board is as much of a topic under pieces as much as it is under board geography. Chinese chess for example has two features that affect how pieces play and some variants use board features to allow some moves that are not possible otherwise, so in this regard the two are intertwined.
A piece moving differently to capture isn't unusual: the pawn for example. The key point here is that the piece only moves a certain way to capture and would not do so otherwise. This then gives us the two general additions: 1) the piece can also capture with its non-capture moves but has additional moves when it captures, 2) the piece has different moves for non-capture and capture.
Next factor is the direction the piece moves in. On a standard square board there are four main directions: left, right, forward and backward. Board geometry allows more directions than these but in practice these four directions define whether a piece can move a certain way, for example Betza's menagerie is filled with pieces defined this way. Tony Paletta made variants where pieces move depending on destination, and a piece can change movement depending on where it stands (commonly through squares of a color on a checkerboard).
In the hopper essay I have talked about conditional hoppers, though the condition in that case is in whether or not a hurdle is involved. Upon further review conditional hopping is more likely to make a piece a hybrid. In a broad sense this section may be about conditional movement, but convention has it that movement types depending on the piece's state are independent of each other and so we need not get too hung up on classifying such pieces; it is a better exercise to classify the conditions needed instead.
Morphin' Like Morphy
Pieces are expected to stay the same in chess. For these kinds of pieces let's disregard that for a bit and give them the ability to change. Thus follows the first question, into what? Conventionally a piece an only change to a piece that exists in the game; you cannot introduce a piece that should not exist. Next thing to consider is what triggers the change: a capture? being at a certain part of the board? merely moving?
Parton created a variant that uses the morph cycle of pawn-knight-bishop-rook-queen-king which may have been the standard cycle for a while(not willing to pin Parton as the source yet).
The other way is for a capturing piece to change into the piece it captured, not to be confused with the capturing piece taking in its properties. In the former a rook capturing a bishop becomes a bishop, in the latter the same situation nets a queen.
In any case, the effects kick in after the change is complete
The Piece Moveth, the Piece Taketh
Pieces usually take as they move, that is their capture moves are the same as their normal moves. Displacement capture is taken for granted in chess that changing it is enough to make it a variant.
Must a piece move to capture? An early variant with a different capture is Rifle Chess, where pieces capture enemies in their range without having to move,
The first serious attempt at playing with more than one capture property is Ultima, giving everyone in the back row who isn't the king other means of capture, seven in total. Later variations trying to improve on Ultima added extra moves, like Rococo replacing some captures with three new ones. Having pieces move the same but take differently doesn't play like chess but is still chess-like albeit with a lot of queen lines.
A capture that has seen some use is that of jumping over pieces to capture them, analogous to checkers. There's even a variant dedicated to it, reminiscent of earlier variant Dynamo Chess.
Wait, What About the Pawn?
Most chess piece taxonomies label the pawn as a separate thing and sometimes add a sort of separate class of pieces based on pawns. This classification assumes variants that are meant to be played, but a pawn is still a piece anyway. A pawn is a sniper as the only piece to move different in a capture or non-capture, a short rider on the basis of its two-step first move but ultimately hops. Only pawns can promote but if a pawn is given enough power this can be done away with.
In the end, a piece is a pawn by virtue of being the first line of defense in the starting position, expendable but nonetheless a vital part of the king's army.
