how many complex solutions does this equation have?

Question

Can someone explain me how to analyse and see the total number of complex solutions for this equation:

Answer 1

A better approach.

Clearly $ z = 0$ is one solution. Otherwise, multiply by $ z $ to obtain

$$ z^n = i | z |^2 $$

Plainly, unless $n=2$ we must have $|z| = 1 $, so we have

$$ z^n= i $$

And therefore $z$ is either $0$ or one of the $n$ roots of $i$.

If $n=2$, then, using real and imaginary parts,

$$ x+iy = i(x-iy) $$

So every multiple of $1 + i$ is a solution.