Find set of $z \in \mathbb{C}$ when $z^4\in \mathbb{R}$

Question

To prove the question in my title. Would my proof below be sufficient. I saw a proof using demoivres formula but was wondering if my solution works.

My attempt

$$(a+ib)^4=a^4-6a^2b^2+b^4+i(4a^3b-4ab^3)$$

So we want $4a^3b=4ab^3$

Which implies that $a^2=b^2$ makes this true.

Answer 1

It means any member of $\mathbb C$ in the form $a+ia$ will make the desired set . Also you can take $0$, $a$ and $ia$ forms.