What's bad about left $\mathbb{H}$-modules? Can you give me non-trivial examples of propositions that can be formulated for every left $k$-module, hold whenever $k$ is a field, but do not hold when $k = \mathbb{H}$ or, more generally, need not hold when $k$ is a division ring (thanks, Bruno Stonek!) which is not a field?
I'm asking because in the theory of vector bundles $\mathbb{H}$-bundles are usually considered alongside those over $\mathbb{R}$ and $\mathbb{C}$, and I'd like to know what to watch out for in this particular case.

Linear algebra works pretty much the same over $\mathbb H$ as over any field. Multilinear algebra, on the other hand, breaks down in places.
You can see this already in the fact that when $V$ and $W$ are left $\mathbb H$-modules, the set of $\mathbb H$-linear maps $\hom_{\mathbb H}(V,W)$ is no longer, in any natural way, an $\mathbb H$-module. As Bruno notes, tensor products also break (unless you are willing to consider also right modules, or bimodules—and then it is you who broke!) 
