<s>[INST] how to solve $|z|^2 + z|z| + 2 = 2i Im(z)$? this is the process I follow:

*

*$z|z|  = 2iImz -|z|^2- 2 $

*$z \sqrt{Re(z)^2 + Im(z)^2} = 2iy - (Re(z)^2 + Im(z)^2) - 2$

*$z\sqrt{x^2 + y^2} =2iy -(x^2+y^2)-2$

*$z=(2iy -(x^2+y^2)-2)/\sqrt{x^2 + y^2}$
then i get: $Im(z)=2y/\sqrt{x^2 + y^2}$ and $Re(z)=(-(x^2+y^2)-2)/\sqrt{x^2 + y^2}$
what should i do next?
 [/INST] You may simplify the notation from the beginning. Call $z=a+bi$ and your equation becomes:
$$a^2+b^2+(a+bi)\sqrt{a^2+b^2}+2=2bi$$ or
$$[a^2+b^2+a\sqrt{a^2+b^2}+2]+[bi\sqrt{a^2+b^2}]=2bi$$
So, you can split it into the real part and the imaginary part.
$$a^2+b^2+a\sqrt{a^2+b^2}+2=0\quad (1)$$
$$b\sqrt{a^2+b^2}=2b \quad (2).$$
From $(2)$ you get $b=0$ or $\sqrt{a^2+b^2}=2$.
Can you finish?
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