I have recently begun reading about non-standard analysis. According to this wikipedia article it is possible to construct an ultrapower $M^I/\mathcal{U}$ from any structure $M$ and index set $I$ with non-principal ultrafilter $\mathcal{U}$ and obtain a transfer principle for it. The hypernaturals, hyperreals, hypercomplexes, etc. are simply the result of choosing $M \in \{\mathbb{N}, \mathbb{R}, \mathbb{C}, \ldots\}$ and $I = \mathbb{N}$.
I was wondering, are there any other significant, meaningful or otherwise interesting choices for $I$? For example is there anything new to be learned from choosing $I = \mathbb{R}$ instead? It seems to me that such a set would be interesting since its elements are essentially equivalence classes of real functions instead of real sequences, yet I couldn't find any mention of such a set online.