If $z$ be a complex number, then one of the solutions of the equation $z^{2} + |z|^{2} = 0$ is …

Let $z = x + iy$, then $|z|^{2} = x^{2} + y^{2}$. So, $(x + iy)^{2} + x^{2} + y^{2} = 0$. $\Rightarrow (2x^{2} - y^{2}) + 2ixy = 0 \Rightarrow 2x^{2} - y^{2} = 0, 2xy = 0$. If $x = 0$, then $-y^{2} = 0 \Rightarrow y = 0$. If $y = 0$, then $2x^{2} = 0 \Rightarrow x = 0$. Non-trivial solution: $x = y = 0$. So we take $z = i\sqrt{2}|x|$ form. $z = 2 + 3i$ satisfies the given equation.