In an early math class, I was shown how all Reals could be constructed from Rationals using a 2-D representation (ex. Real numbers are represented by (a + b \sqrt{2} ) where a & b are Rational). While using the 'lesser' system of Rationals requires a 2-D representation, we can also represent Reals using decimals as single values (a 1-D representation).
If this is the case, then might our current expression for complex numbers behave the same, such that we use the 'lesser' system of Reals to represent complex numbers as 2-D (x + yi), however there could exist a numeric system that represents complex numbers as single values (1-D representation)?
Not really. It can be shown that the reals form the only complete totally ordered field, up to isomorphism.
EDIT: In simpler terms: if you map the complex numbers to a line, you are forced to discard some structure.