i is a number defined to have i² = -1. Everything else follows from that. The complex plane arises naturally as a useful representation of complex numbers, it i

i is a number defined to have i² = -1. Everything else follows from that. The complex plane arises naturally as a useful representation of complex numbers, it is not the starting point. I do not understand at all that stuff about "spiral path between otherwise discontinuous real numbers", but I suspect you may be making things more complicated than they need.