In standard chess, technically a pawn that reaches the back row can promote to any first-rank piece other than the king. In practice, it almost always promotes to a queen, and only very very rarely to a knight. There's no point in promoting to rook or bishop, since the queen contains the moves for both. The choice of promotion would be more interesting if the pawn could promote to any compound of simple first-rank pieces: the queen (bishop+rook), archbishop (bishop+knight), or marshall (rook+knight). Possibly other pieces of similar strength as well.
The idea of allowing pawn promotion to fairy chess pieces not found in the starting array is not new. My idea is for the promotion to be undeclared, and revealed by how the piece is moved. There would therefore be an element of deduction on the part of the opposing player. For example, if the promoted pawn slides orthagonally, the player knows that it has a rook component (a queen or marshall); if it later makes a knight's move, it must be a marshall.
A bit like a particle in quantum mechanics, its move only "collapses" into something definite as it is observed. Until the player commits to a specific move, it could be anything. In other words, the player doesn't have to decide on a promotion as soon as the pawn reaches the back row, but fixes components as he goes. If only one component has been used (for example, an orthagonal move meaning the piece is part rook), the player can still keep his options open and decide whether it's also part knight or part bishop later.
We can say that a pawn that has reached the 8th rank but has not moved yet is unexposed: its nature is not known. One that has made only one type of move is partly exposed. One that has made two types of moves is fully exposed: its move possibilities are fixed, and it is effectively
The problem here is that a pawn that is not yet fully exposed is immensely powerful. If it is unexposed, it threatens all of the squares for any of its potential moves (since it's free to choose any of those moves as a component). If partly exposed, it still threatens all squares for all possible components, because it is still indeterminate between the two unknown components. This is way too powerful, effectively an amazon (rook+bishop+knight): the game Maharajah and the Sepoys pits a single royal (kinglike: unable to move into threatened squares, game lost if it cannot escape capture) amazon on one side against an entire chess army on the other, and the full army only barely outclasses it. To fix this, we can rule that squares are only threatened if the player reveals that they are on his turn (explicit), or they are threatened by a move that has already been exposed (implicit). If the space is not threatened (explicitly or implicitly), the promoted pawn is not allowed to capture a piece that moves there on the next turn (although it may fix a component later that would have allowed it to make that capture if it had been set beforehand).
This rule could also be limited to determining threatened squares only for purposes of giving check (since a king is not allowed to move into check).
Let's add another possible promotion to the mix. A piece with the king's move (minus royal restrictions and castling) is called a "mann", and is a valid basic piece that could be used as a component. The combination of a mann and knight is called a centaur, and while it is short-range and seems weak, it is actually a pretty strong piece in the same range as the previously mentioned compounds (and maybe a bit stronger than the bishop+knight). Mann+bishop and mann+rook can be safely ignored; while perfectly valid pieces in their own right, they are both subsets of the queen and therefore redundant in this case.
The addition of this piece causes some interesting complications. A single-space move, orthagonal or diagonal, no longer unambiguously reveals a full component (a knightwise move still reveals a knight component on its own). A piece that has made a knight's move and a single-space orthagonal move may be a marshall or a centaur (if it also makes a single-space diagonal move, though, it's clearly a centaur; if it slides more than one space orthagonally, it's clearly a marshall). A piece that has made a single-space diagonal move and a knightwise move could be archbishop or centaur. One that has moved only like a king may be queen or centaur.
Now another type of piece: the bent riders gryphon and aanca. Technically these aren't compound pieces at all: a gryphon moves one space diagonally followed by sliding at least one orthagonally away from the starting space; the aanca swaps these (one orthagonal, multiple diagonal). However, I believe they around the same range of strength, especially in the endgame when the board is relatively cleared out. They also make deductions more complicated, because they include the knight's move, but not exactly. Unlike a knightwise jump, they can be blocked: a gryphon on a diagonally adjacent square, the aanca on an orthagonally adjacent square. An unimpeded knight's move could be any piece but a queen, but if there was a piece in the way, the possibilities narrow.
While it seems like this element of uncertainty would be a sort of "wild card", the game is still actually deterministic. It may not seem so, but it is still a complete information game: the possible moves of every piece are known to both players.
Another way of achieving a "black box" effect of pawn promotions would be for a player who promotes a pawn to determine at the time of promotion which piece it now is, note it down, and hold to that decision. Under this rule, there would be no "declared threat": if the piece can capture with one of its moves, it can, and if the opposing player moves his king to a threatened square, he must be told that he has moved into check and allowed to take back the move. While this is probably easier to understand, it actually does add an incomplete information aspect to the game, and would play very differently. Call this one "Clue Chess".