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.