Any hints towards the answers? I've spent a lot of time with complex numbers, and my answers would be

Quaternions: not profound, C is complete, quirky but useful representation of SO(3)

Inverses: fun fact coincidence

1+i/1-i: not sure what to experiment with here

i^i: gateway to riemann surfaces.

Adding angles: comes out like this, that's the point of exp(i phi)

Unit circle: roots of unity?

Riemann sphere: cool stuff!

Quantum stuff: mathematical physicist here, no need to sell this one!