This is an extended version of my comment. Suppose we stay on the surface $z^2+w^2=1$ but away from the origin. The identity $|z^2+w^2|^2=|z|^4+|w|^4+2Re((z \bar w)^2)$ tells us that the square of $z \bar w$ has negative real part. The set of complex numbers $\zeta$ such that $Re(\zeta^2)<0$ has two connected components: it's the disjoint union of two open sectors. Finally, note that switch from (z,w) to (-z, w) involves going from one component to the other.
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