This question occurred to me earlier today.
I can see that if the field has a unit, then there is an element of multiplicative order $2$, namely $-1$. Thus if there was an isomorphism $(F,+) \cong (F^*,*)$ then the characteristic of the field would have to be divisible by two, so that it must be of characteristic two. It would also have to be an infinite field, otherwise $N= N-1$. So I am looking in particular for some infinite field of characteristic 2, or for some other group invariants that would help.
Another possible lead is that $(F^*, *) \subseteq Aut(F, +)$. Does the exponent of a group impact the exponent of its automorphism group? Or something along those lines.
Other than that I am stuck, however. Does anyone know if this is an accessible problem to me? I only know the basics of field theory, extensions and Galois theory. Does anyone have an answer (or hint) that they would like to share?
Edit: Daniel Fischer answered my question.
What about the case when $R$ is a commutative ring and we are trying to compare $(R,+)$ with the group of units of $R$?