Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc. When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to Borel) states that the monodromy action of a generator of $\pi_1(\Delta^{\times}, o) \simeq \mathbb{Z}$ on the Betti cohomology of a fibre is quasi-unipotent (recall that an operator $T$ is called quasi-unipotent if $T^N$ is unipotent for some $N$).

Both of the proofs which I know sufficiently use the algebraicity of the family.

The first one is due to Grothendieck and is based on the action of an absolute Galois group in étale cohomology. The first step is to choose a number field $K$, on which our family is defined.

The second one is due to Schmid and works for arbitrary polarised variations of Hodge structures. It intensively uses the geometry of polarised period domains and I am not sure if non-polarised period domains share the same properties.

Thus, my questions are:

1)Are there known examples of non-polarisable variations of pure Hodge structures over a punctured disc with non-quasi-unipotent monodromy?

2)If yes, are there any of those of geometric origin? More precisely, is there a known example of family of non-projective Kähler manifolds over a punctured disc, for which the quasi-unipotency theorem fails?