Ordering Complex Numbers

The positive numbers can be ordered, so that everyone can agree which of two positive numbers is larger. So can the negative numbers, fractions, and all real numbers be so ordered. All the sets of numbers can be displayed on a number line, usually increasing up or to the right.
A problem arises though when we attempt to order the set of complex numbers. Complex numbers have real and imaginary components. They are usually displayed on an Argand diagram. The real axis is usually the horizontal axis and the imaginary axis is usually the vertical axis. We can only properly displayed complex numbers using two number lines at right angles. For this reason we cannot order the complex numbers.
We cannot construct an sort of order relation for the complex numbers. To see why this is so, suppose we order the numbers by magnitude.
Consider the numbers -1, 1+i, 1
Obviously -1 <1 in the usual sense. If we order by magnitudes then we would write -1=1<1+i which is obviously wrong.