Formally, recall that any structure $N$ admits a diagonal elementary embedding$N \overset{\Delta}{\to} N^{\mathcal{U}}$ to any ultrapower$N^{\mathcal{U}}$ of $N$.

If $i \colon N \to M$ is an elementary embedding, we say that $N$ is an ultraroot of $M$ if there is some ultrafilter$\mathcal{U}$ on some index set such that there is an isomorphism

Similarly, taking a countably-indexed ultrapower of a slightly-expanded copy (like a nonstandard model generated by throwing in a single infinitesimal) of the reals still yields the hyperreal numbers.

Remarks

Closure under taking ultraroots, along with closure under ultraproducts and elementary embeddings, gives a criterion for detecting elementary classes. See also at Birkhoff's HSP theorem.