Right, but when you're working with the function on C x C, it doesn't really add anything either to arbitrarily pick 0^0 = 1, except for consistency with that o

Right, but when you're working with the function on C x C, it doesn't really add anything either to arbitrarily pick 0^0 = 1, except for consistency with that other function that happens to be written the same way. I'm not really opposed to saying 0^0 = 1; it's the only reasonable choice if we're going to insist on using the same notation for these two functions, and I don't expect that's going to change.