I've had Dave's Short Course on Complex Numbers on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers. I'm frequently asked that question.
This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful. It is a question about what makes them numbers. It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.
Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number.