By an ordered field I mean:
a < b => a + c < b + c
a < b and c > 0 => ac < bc.
So you can't group complex numbers into positive and negative numbers. Square root, along with many other functions, has to become multi-valued .....
A complication with that is that you can't just define i by because -1 now has two square roots and there's no way to distinguish between them.
A more formal definition is to define the complex numbers as the set of all pairs of real numbers (a, b) and deine addition and multiplication by (a, b) + (c, d) = (a + c, b + d) and (a, b) * (c, d) = (ac - bd, ad + bc) respectively. It can then be proved that all the rules of arithmetic hold, that pairs of the form (a, 0) obey the same rules as real numbers and that (0, 1) * (0, 1) = (-1, 0).
If the symbols 1 and i are used to mean (1, 0) and (0, 1) respectively, then (a, b) can be written as (a, 0) * (1, 0) + (b, 0) * (0, 1) = a * 1 + b * i = a + ib.