find the Number of solutions of the equation z^2+|z^2|=0 where z is a commplex number and
|z| has its usual meaning

Infinitely many solutions. Since z^2 + |z^2| can be written as z^2 + z.z` (where z` is conjugate of z) . Therefore z^2 +z.z` = 0 ,Then, z(z + z`) = 0. Now let z = x +iy. Thus it will become (x + iy)(x+ iy + x - iy) = 0.ie (x+ iy)(2x) = 0. Therefore it will be 0 for x (real part of z =0). But y( imaginary part can take any value) . Hence it will be true for every pure imaginary number (real part is 0). Thus, infinitely many solutions.