The problem statement is to solve the quadratic equation $p(w) = zw^{2}+(1\color{red}+z^{2})w+z = 0 $ in $\mathbb{C}$.
For Real numbers it's easy, becouse we have $\Delta$ which works for complex version. But am have't idea how to use for complex equation.
Can you had some issue?
The quadratic formula is just fine. It will tell you the two solutions for $w$ in terms of $z$ since the discriminant will have two square roots (unless it's $0$). The solutions need not be complex conjugates (as in the real case).