Topology of the homogenous polynomials

Question

I want to know the connectedness (say fundamental group for example) of the set $$S=\{(z_1,z_2,z_3)\in\mathbb{C}^3|z_1^2+z_2^2+z_3^2\neq 0\}$$ in the usual metric topology. To approach the problem, I want to study the complement of $S$ at first. It's clear that the zero set of the homogenous polynomial $z_1^2+z_2^2+z_3^2$ is a star-domain and thus being simply connected. Consider this problem in the real space $\mathbb{R}^6$, I believe this star domain must be the union of different dimensional $\mathbb{R}^d$ intersecting at the origin with $d\leq4$. I'm stucked at here and don't know how to proceed forward. Any suggestions will be greatly appreciated.