What do we know about the circumstances (whether having to do with the axioms of set theory or the model itself) under which a field $F$ of hyperreals (=ultrapower of $\mathbb R$ with respect to a non-principal ultrafilter on an infinite set) has this property: For every $x \in F \backslash \mathbb R$ there is a field homomorphism (=monomorphism) of $F$ to itself that fixes $\mathbb R$ and moves $x$?

(It looks to me that saturation is a sufficient condition, even if we replace "homomorphism" with "isomorphism".)