This might appear to be fairly innocuous, but it is actually quite an interesting question. First of all, what do we want from an order?

A (total) order should satisfy the following properties for elements of a set X:

1. For any number a, a ≤ a

2. Transitivity: if a ≤ b and b ≤ c, then a ≤ c

3. Antisymmetric: if a ≤ b and b ≤ a, then a=b

4. Totality: a ≤ b or b ≤ a

Ok so that's fine and all, but the complex numbers form a field as well, so we want the order to interact correctly with the field structure. To make a field (F,+,x) and an order < into an ordered field, we need two more things:

1. If a < b then a + c < b + c

2. If 0 < a and 0 < b, then 0 < ab

Now, you can show that if you want all of these things to happen, you cannot have a total order on the complex numbers. However, people still want to compare complex numbers, so how does one go about this? Well, if we can map the complex numbers to the reals in some sensible way, one can compare real numbers in a straightforward way. So, a lot of the time, comparing complex numbers is done by comparing their absolute values.