Question

Find the number of solutions of $z^2+|z^2|=0.$

Answer 1

$z^2+|z^2|=0.z\ne 0\Rightarrow x^2-y^2+i2xy+x^2+y^2=0\Rightarrow 2x^2+i2xy=0\Rightarrow 2x(x+iy)=0\Rightarrow x=0\text{or}x+iy=0\text{(which is not possible)}\therefore x=0,z\ne 0$

So y can have real value.

Hence infinitely many solutions.