Well, IMO their rotational feature is kind of the accidental, happy and mysterious side-effect that starts from the "simple" computational trick of defining sqr

Well, IMO their rotational feature is kind of the accidental, happy and mysterious side-effect that starts from the "simple" computational trick of defining sqrt(-1) being i, and taking the calcuslations from there. I would philosophically say "Completeness" is a more important property (since now negative roots have a meaning attached to them). But their rotational aspect is perhaps more mind blowing.