This is just scratch on the surface.

* Enter quaternions; things get more profound.

* Investigate why multiplicative inverse of i is same as its additive inverse.

* Experiment with (1+i)/(1-i).

* Explore why i^i is real number.

* Ask why multiplication should become an addition for angles.

* Inquire the significance of the unit circle in the complex plane.

* Think bout Riemann's sphere.

* Understand how all this adds helps wave functions and quantum amplitudes.

i^i isn't anything. Please don't write this. Of the two inputs to the function (w, z) -> w^z = exp(z ln(w)), only z is a complex number, so that bit is OK. The problem is that w is NOT a complex number but a point on a particular Riemann surface, namely: The natural domain of the function ln. That particular Riemann surface looks like an endless spiral staircase. The more grown-up term might be "a helix". When you write informally "w=i", that could mean any of ln(w) = i pi/2, i (2pi + pi/2), i(4pi + pi/2), etc. Incidentally, w^z is then always a real number. However, there's an infinite sequence of those numbers that it could equal.

I suppose that by pure convention, "w=e" is understood as denoting a single unique point on the helix. But extending that convention to w=i starts to look like a recipe for confusion.