(While reading about classification of simple groups, I was wondering if there could be something like simple fields or something... And this thought came in mind)
When we talk about field in abstract algebra of linear algebra,
It really feels like the second binary operation-multiplication is very much dependent the first binary operation-addition.
If there is no counter example to it then it implies that study of field is totally dependent on the group theory.
Also every finite dimensional vector space over $F$ is isomorphic to $F^k$.
Since $(\mathbb R,+)$ and $(\mathbb C,+)$ are isomorphic groups (assuming the axiom of choice), there is a binary operation $\times$ on $\mathbb R$ such that $(\mathbb R ,+,\times)$ is isomorphic to the field of complex numbers. Therefore $\times$ is not the same binary operation as real multiplication.
The real numbers and the complex numbers have isomorphic additive groups because they are both vector soaces of (algebraic, i.e., Hamel) dimension $2^{\aleph_0}$ over the field of rational numbers.