There's a related idea in mathematics, the proof that the real numbers are a vector space over the rational numbers. If you scramble the basis vectors, you obtain an isomorphic vector space, but it is effectively a "permutation" of |R. Of course, vector spaces don't even have multiplication, but one interesting thing is that the proof requires the axiom of choice.

I think that actually constructing a "nontrivial" model of C using the field conception might require choosing a member from each of an infinite family of sets, i.e. it requires applying the axiom of choice, similar to the way you construct R as a vector space.