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!

It is possible to find all roots by an easy observation: The right hand side is always real, which means that $z$ either has to be real or purely imaginary so that $z^4\in\mathbb R$.
Let $z=a+bi,$ wehre  $a,b\in\mathbb R$. If you want to find the real solutions, work with $z=a$. Since you were able to find these, I'll skip them.
In order to find the purely imaginary solutions, set $a=0$ so that $z=bi$. Substituting this into our original equation:
$$(bi)^4=\lvert bi\rvert^2+2\Longrightarrow b^4=b^2+2.$$
This is now a simple quadratic equation (set $u=b^2$) you should be able to solve.
