While I like the geometric content of the rest of your post, I have to quibble on two points:

    * that a number has two (usually distinct) square roots is a theorem, not an axiom.  That is, it's not 'built in' (unlike, say, commutativity of addition), but rather can be proven on the basis of 'built-in' properties.

    * while one may technically say that i and -i have 'opposite signs', it's a good idea not to talk about the sign of a complex number at all (except possibly in the sense that sgn(z) = z/|z|).  It's better to say that the two square roots are opposite; or, if one wants to be more precise, that they are additive inverses.