As usual, since I spent some time writing this up on /r/learnmath, I’d like to share it with you guys. The OP asks:
what does it mean to raise a number to the power of i?
I explain in this comment, but here it is too.
Alright, so, we’re talking about complex numbers, so I’m assuming you know how to plot them in the complex plane and do basic addition/multiplication.
If we want to know what a number like $2^3$ means, it’s pretty easy: you take $2×2×2=8$. Same for $(-3i)^3$ (Wolfram Alpha): it’s $-3i\times -3i\times -3i=-27i$.
Think about what’s happening visually: $2^3$ stretches the vector from the origin which used to point to $2+0i$, but now it points to $8+0i$. The effect on $(-3i)^3$ is more complicated: there’s stretching, and reflection across the origin.
When you mix the two together, say, $(2-3i)^3$ (Wolfram Alpha) which is equal to $-46-9i$, you get stretching, and a reflection across the imaginary axis.
So, here’s where the usefulness of raising a number to an imaginary (or complex) number kicks in: it gets us arbitrary rotations. It just so happens that multiplying a complex number by $e^{\theta i}$ rotates that number $\theta$ radians counter-clockwise around the origin. For example, $3e^{\frac{\pi}{2}i}$ (Wolfram Alpha) is $3i$ since that represents a rotation of $\frac{\pi}{2}$ or $90$ degrees.
You can take any complex number of the form $(a+bi)^i$ and manipulate it to be in the form $(c+di)e^{\theta i}$ (although that’s out of the scope of this post). For example: $(1+i)^i$ (Wolfram Alpha) can be manipulated to become $e^{\frac{\pi}{4}}2^{\frac{1}{2}i}$, which can be rewritten as $e^{\frac{\pi}{4}}e^{\ln(\sqrt{2})i}$ . So we’re taking the point $e^{\frac{\pi}{4}}+0i$, and rotating it $\ln(\sqrt{2})$ (about $0.34$ or $\frac{\pi}{9}$) radians around the origin.
Since $(a+bi)^{c+di} = (a+bi)^c(a+bi)^{di}$, we see that raising a complex number to a complex number represents a certain exponential stretch or compression from the origin, followed by a certain rotation.
tl;dr: You can write $(a+bi)^{c+di}$ to be of the form $(x+yi)e^{\theta i}$, which means take the complex point $x+yi$ and rotate it $\theta$ radians counter-clockwise around the origin of the complex plane.
(Apologies for the rawness of the explanation… I’d really like to clean this up down the road and put in some pictures. Hope someone thinks it’s useful, though!)