<s>[INST] Solve $z^2 + 4|z| + 4 = 0$ in $\mathbb{C}$ 
How can I solve this equation:
  $z^2 + 4|z| + 4 = 0$?

If I rewrite $z$ as $a+ib$, I get 2 real solutions. But they are wrong.
Any help?
My attempt. By letting $z=a+ib$, we have that
$$(a+ib)^2 + 4 \sqrt{a^2+b^2} + 4 = 0 $$
$$ a^2 + 2iab - b^2 + 4 \sqrt{a^2+b^2} + 4 = 0 $$
$$a^2 -b^2 + 4 \sqrt{a^2+b^2} + 4 = 0 $$
Then $2ab = 0 $.
So, for $a = 0 \implies z=2$ and $b = 0\implies z=-2$
 [/INST] Hint: $z^2 + 4|z| + 4 = 0$ iff $z^2 =-4 |z| - 4 \le 0$ and so $z=bi$ for some $b$.
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