That is not actually the way to define e^x for complex numbers. There are a few ways you can do so. The most natural way is as the unique solution to the differential equation

df/dx = f

The solutions are e^x + const; requiring f(0) = 1 gives you the choice const = 0.

Another way to define it is as the infinite series

e^x = \sum_{i=0}^inf x^i/i!

It is pretty easy to show this definition is equivalent to the previous one. A third way is as

lim_{n -> inf} (1 + x/n)^n

This definition comes from the intuition that e^x represents the limit of continually compounding interest. As before, it is pretty easy to show that it is equivalent to the previous two.

In any case, all these definitions extend directly to complex values of x. The fact that

e^{ix} = cos(x) + i sin(x)

holds (and hence Euler's identity holds) is a consequence of a more natural definition, not the typical definition.

Further evidence that e^{ix} = cos(x) + i sin(x) is natural is that it fits in nicely with power series representations. If you define cos(x), sin(x), and e^x by their power series centered at 0 then it's straightforward to see that substituting ix into the power series for e^x yields the sum of the power series for cos(x) and i*sin(x) (as long as you accept that theorems about absolute convergence and rearranging terms extend to the complex numbers).

The lecture in my Calculus IV course where we walked through that derivation is literally my only memorable math lecture in university. It just so beautifully ties so many different concepts together!

i x is x rotated left by pi/2

df/dx = f

df(ix)/dx = i f (ix)

so the "velocity" is always orthogonal to the "position". Thus the solution to this equation in the complex numbers has to be a rotation.

The derivative of e^ix is ie^ix. Multiplication by i is equivalent to rotation by 90 degrees. The derivative of velocity is acceleration. Acceleration perpendicular to velocity is circular motion.

With base 2 the motion is not circular.

All these pieces (and an initial value problem differential equation) come together to explain the identity.

The surprise is that the definition you gave in 2. fully agrees with the definition of multiplication that comes naturally from the cartesian version of complex numbers: z = x + bi -> (x1 + b1i)(x2 + b2i) = x1x2 + x1b2i + x2b1i - b1b2

Vector multiplication has two very different forms (dot and cross), so multiplication (and by extension exponentiation) working out smoothly is neat.

Complex exponentiation is not defined arbitrarily to make Euler identity true. The Euler identity is a consequence.

I love the Euler identity in its typical form, but I think it is more surprising when you replace e and pi by their (approximate) numerical values and i by sqrt(-1)

> Why is the base e? Why isn't it 2, like this

One of the many things that makes e special is that e^x is the only function that is its own derivative. One of the things that make sin and cos special is that they are each other's derivatives (except for a change of sign in the case of dcos/dx). This doesn't prove anything on its own, but it suggests that there might be some relationship between e and the trig functions, and Euler's equation proves that indeed there is.

I suspect it would spiral off the unit circle.