For the function $F: \mathbb{C}^3\to \mathbb{C}$, $(x,y,z)\mapsto x^2+y^2+z^2$, $0$ is a regular value. So in particular, $F^{-1}(0)\subset \mathbb{C}^3$ is an embedded submanifold (of dimension 2 over $\mathbb{C}$ or dimension 4 over $\mathbb{R}$). I am having trouble however visioning what set looks like. My end goal is to put a CW structure on $q(\text{im}(F))$ where $q: \mathbb{C}^3\to \mathbb{C} P^2$ is the quotient map but I would like to understand what this submanifold looks like first. I have tried writing this out in terms of the real and complex parts of the three complex numbers. This gives relations $$ \sum_{i=1}^3 Re(z_i)=\sum_{i=1}^3 Im(z_i), \; \text{ and }\sum_{i=1}^3 Im(z_i)Re(z_i)=0. $$ But I do not see how this would help. Its also clear that if $F(x,y,z)=0$, then $F(kx,ky,kz)=0$ for any $k\in \mathbb{C}$ but I don't know how to utilize this either.
I would really appreciate if someone could give me some input on how to visualize this set or how I would go about putting a CW-structure on $q(\text{im}(F))$.