why imaginary numbers have a a different meaning than $R^2$ if they are semantically equivalent?

Question

why imaginary numbers have a a different meaning than $R^2$ if they are semantically equivalent? regardless of the historical perspective, we know that ther is no semantic difference between them

Answer 1

They are the same as sets and they are isomorphic as (real) vector spaces, however $\mathbb{C}$ is also endowed with a multiplicative structure with respect to which it is a field. Of course we could make $\mathbb{R}^2$ a field as well by definining an appropriate binary multiplication operator but there are many ways to do this.

This lack of multiplicative structure on $\mathbb{R}^2$ by default is what makes holomorphy (differentiability as a map from $\mathbb{C}$ to $\mathbb{C}$) a stronger property than being a differentiable map from $\mathbb{R}^2$ to $\mathbb{C}$ for example.