Complex solutions of $z^4 = |z|^2 +2$ I'm struggling with the equation in the subject, I managed to find the real solutions ($\pm\sqrt{2}$) by setting $u = z^2$ and I know from Wolfram Alpha that the other two complex solutions should be $\pm i\sqrt{2}$ and I obviouvsly understand why those works but I'm unable to get the correct procedure to find them.
I suspect that there is something around equations with modulus in them that I don't yet know.
Could you point me to the right direction? Thanks!

If $z^4=|z|^2+2$, then$$|z|^4=\left|z^4\right|=\left||z|^2+2\right|=|z|^2+2,$$and therefore $|z|=\pm\sqrt2$.
So, $z=\sqrt2e^{i\theta}$ for some $\theta\in\Bbb R$. And$$z^4=|z|^2+2\iff4e^{4i\theta}=4.$$Can you take it from here?
