For starters, we'll want to check parity. Every bishop-like move stays on the same coloured square, and every knight-like move switches colours. This means that we visit two light squares and then two dark ones, so it once we have reached 64 squares, 32 of them will have been dark, which is exactly right.
With this out of the way, we need to find a strategy. (Brute forcing the possibilities would take too long otherwise.)
The most beautiful strategy would be to find some compact shapes which can be filled at once, figure out their allowed connections, and tile the entire board with them. As an example shape, starting as a knight in the corner of an otherwise empty 3x3 square, it's easy to visit the entire 3x3 square, ending as a knight in the middle spot, allowing for 8 different directions to continue in.
If we try this approach, we'll find out it definitely works for the "just some path" case, but we'll just as soon notice that if we want to find a closed path, we're going to need a more powerful tool.
So this time we'll start by drawing two lines from the chameleon in the corner:
- The forward path, which starts with a knight's move, and can make long bishop's moves unless any of the skipped squares were already visited by the path itself, and
- The return path, which starts with a bishop's move, and can make long bishop's moves only if all of the skipped squares were already visited by that path itself.
With those restrictions, all the long bishop's moves will only jump over squares that are unvisited at the time of the jump.
Turns out that
It's not all that hard to find a closed path this way: