Yup, calculus works in the complex field as well =)
treat "i" as just any algebraic constant.
There's a fundamental flaw to this question.
Remember you don't have a definition for e^z where z is complex. Which means that you will either need an explicit defintion for e^z to define the derivative or you can define exp(z) to be a function such that d/dz exp(z) = exp(z)
The first approach will probably mean defining e^z in terms of a power series, but then one can just check euler's identity by adding the power series.
the second approach will require you to prove that such a function in fact exists and that it agrees with what we know to be e^z when z is real.
Other approaches will require more machinery.
just to illustrate, the function f(x) = exp(-1/x) for x > 0 and 0 when x <= 0 is smooth (infinitely differentiable) when considered over the real numbers. but if you consider it as a complex number function then it has an essential discontinuity at 0