Solve $z^2 + 4|z| + 4 = 0$ in $\mathbb{C}$

Question

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 $b = 0\implies z=-2$

Answer 1

Hint: $z^2 + 4|z| + 4 = 0$ iff $z^2 =-4 |z| - 4 \le 0$ and so $z=bi$ for some $b$.