TITLE: Solving complex numbers equation $z^3 = \overline{z} $
QUESTION [6 upvotes]: We have the following equation:
$$z^3 = \overline{z} $$
I set z to be $z = a + ib$ and since I know that $ \overline{z} = a - ib$. I was trying to solve it by opening the left side of the equation.
$$ z^3 = (a+ib)^3 \Rightarrow $$
$$ [a^2+b^2+i(ab + ba)](a+ib) \Rightarrow $$
$$ a^3 - b^2a - 2b^2a + i (2a^2b + b^2a - b^3) $$
but this is where I got so far and I'm not sure how continue and if my solution so far is even the right way to solve it.

REPLY [2 votes]: Multiplying both sides by $z$ (that does not introduce new solutions), $z^4=|z|^2$ is a real number.
Then by the imaginary part,
$$4a^3b-4ab^3=0,$$
$$a=0\lor b=0\lor a^2=b^2.$$
The real part gives,
$$a^4-6a^2b^2+b^4=a^2+b^2$$
which simplifies to 
$$b^4=b^2\lor a^4=a^2\lor-4a^4=2a^2,$$
and it is an easy matter to list the five solutions,
$$\color{green}{(0,0),(1,0),(-1,0),(0,1),(0,-1)}.$$