Let $z = a + ib , z^* = a - ib$
I need to find all possible solutions to $2z=(z^*)^2$
$(z^*)^2 = (a^2 - b^2) -i2ab$
$2z = 2a + i2b$
$\implies 2a + i2b = a^2 -b^2 -ia2b$ $\implies a^2 - b^2 -i2ab - 2a - i2b = 0$ $\implies a(a -i2b - 2) - b(b + i2) = 0$
I found solution to the above equation $z = 0$. However, I do not know how to find the rest of the solutions from here.