The question at hand is stated in the title.
Typical proofs that I have read involves taking a complex number i, then reaching some contradiction with i2 > 0.
I feel like these proofs bypass the problem of what it means to define an order in the complex field. In other words, what does it mean to have 1 complex number larger/smaller than the other?
Intuitively, taking the norm is the obvious way to compare, but that violates 'if x>y, y-x < 0'. Nonetheless, the cause to my confusion is there seems to be many ways that an 'order' can be defined, and until each one of them is explored, one cannot prove if a complex field can or cannot be ordered.
Keen to hear your comments on my amateurish view.