$\mathbb{R}$ with usual $+$, but not multiplication

Question

I want to find a field whose elements are the real numbers, whose addition is defined the usual way, but which is not isomorphic to $\mathbb{R}$ because of a different multiplicative group. Does this engender a contradiction, or can one construct such a thing?

Thanks!

Answer 1

Assuming the axiom of choice, the additive group of $\Bbb R$ is isomorphic to the additive group of $\Bbb C$, so we can take this isomorphism and pull back the multiplication from $\Bbb C$.