<s>[INST] Solve complex equations with absolute value $|z|^2+z|z|+\bar{z} = 0$ I literally have no idea on how to solve equations like this:
$$|z|^2+z|z|+\bar{z} = 0$$
Even if I try to rewrite it as 
$$(x^2+y^2)+(x+iy)\sqrt{x^2+y^2}+(x-iy) = 0$$
how should I proceed?
 [/INST] Looking at imaginary parts:
$$y\sqrt{x^2+y^2}-y=0,$$
so $y=0$ or $\sqrt{x^2+y^2}-1=0$. 
In the case $y=0$, you then just have the equation $2x^2+x=0$, so $x=0$ or $-\frac{1}{2}$, so $z=0$ or $z=-\frac{1}{2}$. But we find if $z=-\frac{1}{2}$, the real part is not zero, so we reject this solution.
In the case $\sqrt{x^2+y^2}=1$, we have $|z|=1$, so the equation becomes $1+z+\overline{z}=0$, which implies $x=-\frac{1}{2}$, so $y=\pm\sqrt{1-x^2}=\pm\frac{\sqrt{3}}{2}$ and $z=-\frac{1}{2}\pm\frac{\sqrt{3}}{2}i$.
Therefore, the solutions are $z=0,-\frac{1}{2}\pm\frac{\sqrt{3}}{2}i$.
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