Are the complex numbers best represented in 1 or 2 dimensions?

Question

We note that all equations of the type $x+a=b$ can be solved using numbers in one dimension (such as all real numbers), and adding another composition rule, multiplication, we note that two composition rules in a similar way lead us to a two-dimensional system $(a,b)=a+bi$.

I was speculating about a hypothetical third composition rule that, for solution of similar simple equations, would require a three-dimensional number field. It then occurred to me that complex numbers can be viewed as having only one dimension, since all numbers can be generated by numbers with the imaginary components, since $a+bi=-(1*i)*(ai)+bi$.

Do you see complex numbers as one or two-dimensional? (A bonus question is whether you have ever encountered such a third composition rule)

I realize the question is somewhat vague, but it is interesting to me.

Answer 1

When doing complex analysis "for real", no pun intended, it's usually most helpful to think of $\mathbb{C}$ as a one-dimensional (complex vector space or algebra), but when first learning about complex numbers, it's helpful to view it as a two-dimensional (real vector space), since operations on real numbers feel more familiar to most.

There are situations where both viewpoints are helpful and contribute different things. For example, when doing complex analysis in $\mathbb{C}^n$, the tangent space to a domain $\Omega \subset \mathbb{C}^n$ is a $2n-1$-real dimensional object, and sometimes you need to understand and consider all the "real directions" in the tangent space, but most of the time, it's the biggest complex subspace of the tangent space (which turns out to be a $n-1$- complex dimensional or $2n-2$-real dimensional thing) that carries most of the information.