What do we know about the circumstances (whether having to do with the axioms of set theory or the model itself) under which a non-standard model $F$ of analysis 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".)

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".)