Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some radius $\delta$ around $0 \in \mathbb{C}$ so that $0$ is the only critical value in $D$, and we can choose a ball $B$ of some radius $\epsilon$ around $0 \in \mathbb{C}^{n+1}$ so that $0$ is the only critical point in $B$.

It is known that restricting $f$ to $B^{*}=B \setminus f^{-1}(0) \cap B$ gives a fibration over $D^{*}=D \setminus 0$, the so-called Milnor fibration, and it is known furthermore that any fibre of $f:B^{*} \rightarrow D^{*}$ has the homotopy type of a bouquet of $\mu$ spheres $S^{n}$, so in particular the only interesting cohomology of the fibre is in degree $n$, where it has rank $\mu$. The number $\mu$ here is known as the Milnor number of singularity of $f^{-1}(0)$ at $0$.

Choosing a generator for $\pi_{1}(D^{*},p)$ (say a counterclockwise circle starting at a basepoint $p$), we get a monodromy automorphism $T : H^{n}(f^{-1}(p),Z) \rightarrow H^{n}(f^{-1}(p),Z)$. There are some cases when this monodromy automorphism is understood. For example, if the original polynomial is "quasi-homogeneous", $T$ is known to be semisimple and the eigenvalues can be read off from the quasi-homogeneous weights.

I would like a simple example of $f$ (and maybe small $\mu$) where $T$ is not semisimple.