Mapping $2$ kinds of operators?

Question

So $z$ from complex analysis can be mapped as:

$$ z \to \hat z = \begin{pmatrix} \Re z \\ \Im z \end{pmatrix}$$

Now the $+$ in complex analysis seems to be the same operation as the $+$ in linear algebra. So can one map the pluses to each other?

$$ + \to +$$

It seems like a good redundancy of symbols. I clearly can't be the first one to think of mapping operators? This seems like a natural question to ask. What is the application of this kind of maths?

P.S: This was inspired by Contour integration in matrices?

Answer 1

You have rediscovered the fact that $\mathbb C$ is a 2-dimensional vector space over $\mathbb R$. You want to learn about linear algebra.

Answer 2

Not sure what you mean by "application for this kind of maths" but operators and the sets they operate on get mapped all the time. The general term is isomorphism if the underlying elements are mapped $1$-to-$1$ (as in your example), or more generally homomorphism if many-to-$1$ maps are allowed. The concept can be even further abstracted as morphism.