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.

There are lots of results on the topology of the Milnor fibre. For example if $f : (\mathbb{C}^n,0) \to (\mathbb{C},0)$ is a holomorphic map germ with an isolated critical point at $0 \in \mathbb{C}^n$ then the Milnor fibre is homotopy equivalent to the bouquet of $\mu$-spheres, where $\mu$ denotes the Milnor number. The Milnor number is given by the absolute value of the Poincaré-Hopf index of the gradient vector field $\nabla f$ at $0 \in \mathbb{C}^n.$

Take a look at the introduction to this paper for a little more detail and some references. Take a look at this paper for the non-isolated case. Both papers include information about the links.