Extended Squares Chess has the regular 8×8 board and regular pieces with certain squares extending into other squares. This means that pieces in the extended areas can use or move to any of the unoccupied spaces that make up these areas before making their move. An augmented or extended move you could say. The exception is a piece in front of another of a player's own on an extended area, pawn in front of bishop at start for example where the bishop cannot move to the space in front of the pawn straight away because the pawn impedes it's path. The extended areas can function as one square or four squares but if one or more of the four squares are occupied by one player and the opponent moves to any unoccupied square within that extended area they can choose any one of the pieces to capture. But then taking one could mean being taken by one of the remaining next move or any other of the opponent's pieces in a position to move into that extended area. So to start with one half of the pawns can move one space forward, the other half one space diagonally forward before making an actual orthodox move and two squares forward first move still applies. Knights being able to move through other pieces can move to unoccupied spaces in extended areas straight away before moving the conventional way. Bishops have to wait till pawns on the extended area have moved before making use of it.
Above is a randomized variation where more of the squares are extended. When pieces move into an extended area they can use any of the unoccupied spaces in that area to make their next move unless they are impeded by other pieces. Knights can use any unoccupied space in extended areas.
Another variation above with patterned board of yellow cross-shaped extended areas. As stated above pieces moving into extended areas can move to any unoccupied space within that area if unimpeded by other pieces. Knights cannot be impeded. If more than one piece of the same are in an extended area an opposing piece moving into that area can choose which piece to capture.