How do I determine all complex solutions of $|z|z=-i(\bar{z})$, for all $z\in \mathbb{C}$ 
Let $z\in \mathbb{C}$. How do I determine all complex solutions of $|z|z=-i(\bar{z})$?

My approach: We can see that $$|x+yi|(x+yi)=y-ix$$ Then I came up with the real part $$-y^2+x^2+x^4-y^4$$ and the imaginary part $$2ixy+x^2(2iyx)+y^2(2xiy)$$
I don't know how to go on and I think that is wrong anyway.I would appreciate every attempt to help!

If $|z|=1$, then $\overline{z} = 1/z$ and the equation becomes
$$
z = -i/z,
\\
z^2 = -i
$$
with two solutions
$$
z = \frac{-1+i}{\sqrt{2}},\qquad z=\frac{1-i}{\sqrt{2}}
$$
