There's actually a reasonably intuitive reason this one works. (At least for me, the Taylor expansions just look like a crazy coincidence). If you try to think about complex powers using the normal definition of exponentiation, you'll just get a headache. Multiply e by itself i times?
However, there's another way to think about exponentiation. An exponential function's derivative with respect to the exponent is proportional to the value of the function, with the base determining the constant of proportionality. This fact can be intuited. Consider the following non-rigorous observation: x^(n+1) = x * (x^n), so x^(n+1) - (x^n) = (x-1) (x^n).
Now think about y = e^x, where d/dx (e^x) = (e^x). In words, the growth rate of y is the current value of y. Start with e^0=1, so the growth rate is 1. If we increase x by dx, we increase y by dx. If we increase x by idx, we increase y by idx. So, if we draw a vector from the origin to the position of y on the complex plane, the vector will be in the real direction, and y will be increasing in the complex direction, perpendicular to said vector. So the absolute value of y isn't increasing.
Now, in general, whatever the value of y, an increase of x by idx will increase y by iydx. Multiplication by i has the effect of a ninety-degree rotation to the left in the complex plane; try some examples if you don't accept this. So, whatever the current value of y, a small increase in x will move y perpendicular to the vector pointing to y. This results in a circular motion on the complex plane, as long as the i component of x is increased. When y gets halfway through this circle, it's back on the real axis, at -1. When does it get there? Well, since y has moved in a circle of radius 1, it has traveled a distance (not displacement) of magnitude pi to get through a semicircle. Since |dy/dx| started as one, and hasn't changed, that means the absolute value of x has increased by pi. Since we have only increased the imaginary component of x, then x must be i * pi. So y = -1 when x = i * pi. e^(i * pi)= -1; e^(i * pi) +1 = 0.