I think you will be interested in reading about the so-called Milnor map, and the so-called Minor Fibration Theorem. 
The standard reference is John Milnor's book "*Singular points of complex hypersurfaces*" from 1968.
Although people ususally assume that the hypersurface has an isolated singularity. That means that the singularity has finite multiplicity. The non-isolated case (where the singular set is a smooth variety) is much more complicated.
Nevertheless, as a starting point, I recommend looking at the Milnor Map and the Milnor Fibration Theorem. 

Donu's right about the link. You usually consider a function germ $f : (\mathbb{C}^n,0) \to (\mathbb{C},0).$ There are lots of results on the topology of the link. For example, it's homotopy equivalent to a bouquet (wedge product) of $\mu$-spheres where $\mu$ is the Milnor number of $f$; which is finite if and only if $f$ has an isolated critical point at $0 \in \mathbb{C}^n$. In the complex case, the Milnor number is the absolute value of the Poicaré-Hopf index of the gradient vector field $\nabla f$ at $0 \in \mathbb{C}^n.$