This may be more of a comment or opinion.

A 2-Hilbert space is a Hilbert space on two levels: It's enriched in Hilbert spaces (morphisms) and it's a categorification of a Hilbert space in the sense of having biproducts for addition, Deligne-product with $\mathrm{Vect}$ for scalar multiplication and dual structure for the sesquilinear form. Which of those levels do you want to have infinite dimensional? The first one shouldn't be a big deal, you just allow for infinite dimensional Hilbert spaces as $\mathrm{hom}$-spaces.

The second one is the harder one. I personally think that it is no big deal "formally", it just amounts to dropping the requirement that the 2-Hilbert space be finitely semisimple. The question is, will you find *interesting* examples that are actually any use? You have the same in vector spaces. I can cook up tons of infinite dimensional vector spaces, but how do they become interesting and tractable at the same time? By adding some extra structure, say as a Banach space, a $C^*$-algebra, an $\mathrm{L}^2$-space over some measurable space or so. In that sense, I'd say that there already is a theory of "dull" infinite dimensional 2-Hilbert spaces, but people are thinking about interesting ones with extra structure, such as having a measure on objects.

As for your final question, I'd guess it will depend on the context of what you're doing. Are you considering some $n$-dimensional moduli space ($0 < n < \infty$)? Smooth groupoids? Then the approach with measurable categories may suffice. If your additional structure is different, a different variant might be needed.

This post imported from StackExchange MathOverflow at 2014-12-13 14:17 (UTC), posted by SE-user Turion