$z^3 + 3i \bar{z} = 0$

first off it should be clear that $z=0$ is a solution

Now assume $z\neq 0$

rewrite the equation as

$r^3 e^{i3x}+3e^{i\pi/2}re^{-x}=0$

I leave it to you to show that $r=\sqrt{3}$

Given that, you can remove $r$ from the equation and solve for all of the $x$

I leave it to you to show that they are

$x \in \{5,~7,~9,~13,~15\} \cdot \dfrac \pi 8$