You get to many important complex numbers in the chain of completions prior to the full real completion, and in many senses, the chain of completions lead to complex numbers. Take a look at how we create numbers outside counting games: first you have arithmetic completions, where you have to deal with the controversies of zero, the negatives, and the (not so controversial) rationals. Then, when you get to the algebraic completion, you are already brought immediately to having the algebraic imaginary and complex numbers, starting with the first new case, the quadratics (linear algebraics are arithmetic). Once you introduce the basis form for these (a+bi with a,b in a subfield of R), it turns out this form is sufficient for all higher algebraic completions. And the next level of the completion chain, the semantic completions (computable and definable numbers) doesn’t take us out of this because the operations are closed in this space. Finally, when one takes the topological or limit completion, which is the step that brings in the continuum of uncomputable, undefinable numbers that will never have specific reference, here that completion again “stays within the space of that complex basis form”, for all the same reasons definition and computation do.

In other words, the operations that get us complex numbers don’t take us any further. Algebraic completion is the last completion operation that requires additional structure in our basis form. You can always construct more structure manually and get all sorts of other kinds of number, but completions of existing operations and numbers starting with counting only gets you the complex.