I'm no an expert in complex numbers, or quantum theory or a heavy user of them. But I am a computer programmer. To a computer programmer, the fundamental diff

I'm no an expert in complex numbers, or quantum theory or a heavy user of them. But I am a computer programmer. To a computer programmer, the fundamental difference between complex numbers is they can express infinite repetition succinctly. Now I try to write down what that means precisely, it's hard. An example of the effect is the polynomial for sin() is infinite, or you can express it as sin(x) = (e^(ix) - e^(-ix))/2i using complex numbers. To a computer programmer who makes his living from writing down formulas, the difference between having to write an infinite amount of code to express a concept or just use 20 characters above is profound. Different people find their profundity in different places, I guess - but this the sort of thing that drives us programmers to create entire new computer languages. That's not to say the primary observation I see being made here is wrong. That observation is that there complex numbers bring nothing really new to the table. You can do everything they do in other ways, say with matrices and a few extra rules. And indeed, mathematicians have come up with numerous other ways of expressing iteration, ∫ and Σ springing to mind. But nonetheless, the hyperoperations (counting, addition, multiplication, exponentiation, ...) are special. They are most heavily used mathematical operations, by a huge margin. They have one job - to capture the operation we programming nerds call iteration. But when restricted to real numbers, (I'm no mathematician, so this is conjecture), they can't capture self similarity. The result of function expressed as a finite number of hyperoperations operating on reals always flys off to infinity, or asymptotically approaches a constant, or is undefined over part of it's domain. When you add complex numbers you get another possibility: a possibly intricate pattern repeated infinitely. So yes, complex numbers are basically just a structure containing two real numbers and some modified behaviour, and these is nothing special about that. But there is something profound in the way they allow the planets favourite mathematical operations to finitely express infinitely more behaviours - with no more lexical overhead.