 Solution

Write $z=x+iy$, so the real and imaginary components are $u=x^2$ and $v=y^2$. The Cauchy-Riemann equations require $u_x=v_y$, namely $2x=2y$; the other equation is $u_y=-v_x=0$. The first relation fails off the line $x=y$, so the function is not holomorphic on $\mathbb{C}$. It is complex differentiable at points of that line, but a line contains no open neighborhood on which the equations hold.