Questions in respect to rotation post-PCA have been answered before -> its all in the hands of the researcher... Same answer to the question if rotation (orthogonal or not) makes sense before plugging the components into a cluster analysis?
If the PCA is only being done as a pre-processing step and the diagnostics of the PCA are ignored and the PCA is not used to interpret the resulting cluster analysis then it makes no sense to rotate the PCA. However, rotating it isn't going to invalidate the cluster analysis in any way.
However, if it is important to interpret the clusters then it is often useful to rotate the components as you can then interpret the loadings and the other diagnostics and decide how many components to use in the cluster analysis. Further, when you go down this path you can quickly intepret the clusters by evaluating how they differ in terms of the means of the components.
I would not be considering any non-orthogonal rotations. Such rotations are typically only used in situations where researchers are keenly interested in the interpretation of the PCA and have various theoretical beliefs regarding the underlying structure of the data (e.g., if developing psychometric tests). In situations where it is just a pre-processing step non-orthogonal rotations become problematic (e.g., perhaps they will create a structure in the data that is then identified by the cluster analysis). At the risk of making up a number, 99.99% of PCA applications by experienced researchers that are rotated do so using Varimax.