You are mixing up a couple of things. This "dimension" you're talking about is probably the dimension of a vector space. The real numbers, as a vector space over the field of real numbers (recall that vector spaces are defined over fields of numbers), have dimension 1. However, you can consider the vector space of real numbers over the field of rational numbers. What is the dimension of this vector space? Well, the dimension of a (finite-dimensional) vector space is given by the number of elements in a basis for it. So how do we go about finding a basis for R over Q? A basis for R over R consists solely of the number "1", since given a real number "x" there exists an element from the field (in this case R), namely x itself, such that multiplying it by 1 will give you the number x. Yes, this sounds obvious, but that's how you prove that R over R has dimension 1. So, going back to R over Q, we see that "1" cannot be a basis, since if we pick, say, pi, there is no rational that we multiply 1 by to give pi. Moreover, there cannot be any finite number of real numbers that would make up a basis for R over Q, since then R would be countable (look this up if you don't know what it is). So the dimensionality of a vector space depends on which field you're considering. By the way, the vector space of complex numbers over the reals has dimension 2 (a basis is {1, i}), but over the complex numbers it has dimension 1. So, there's an "extra" dimension only if you consider it over the reals.

C (complex numbers) is a field, just like R (real numbers). It's an algebraically closed field, unlike R, so in some sense it's actually the most natural set to call "numbers".

https://en.wikipedia.org/wiki/Algebraically_closed_field

I'm not trying to argue for R being more a set than C.

In my head a one dimensional thing like R is fundamentally different from a multidimensional thing like C.

Well I was answering the question about why is C considered numbers - it's a field, hence elements of C behave exactly like numbers. Moreover, it's an 'algebraic closure' of R, hence a more natural choice for "all numbers", and it's a maximal one at that (i.e. quaternions and such are no longer fields and don't really behave like numbers, while C still does).

Edit: FWIW, I consider the terminology of 'real' vs 'imaginary' completely stupid and misleading. This terminology didn't really make it to other languages, e.g. in Russian it's 'material' vs 'complex' numbers, but they don't use the term 'imaginary'.

We do: вещественная часть и мнимая часть.

Oh yeah, forgot about that. Well, then Russian terms are as stupid.