[nblgfesi] Gryphon variations

The gryphon is a neat fairy chess piece functioning kind of like a fat rook.  Suppose a gryphon sits at (0,0).  The unmodified original piece can move to (1,n) where n>0, plus reflections.  Here are some possible variations of it:

Remove the constraint that (1,1) must be empty, giving it jumping ability like a knight.

Allow it to also move to (1,0).  This makes its attack area shaped like a hash symbol.

We can also imagine an analogous fat bishop.  It is not colorbound; in fact it alternates colors each move like a knight.

The crooked bishop is also somewhat like a fat rook, attacking not just the rank or file it is on, but also the adjacent files.  We could imagine a hybrid: a colorbound gryphon.  Several possibilities of, what squares must be empty to permit the colorbound gryphon to progress?  One possibility is it has crazy jumping ability, kind of like a nightrider, actually dabbaba-rider.