I know the traditional way to define complex numbers is to start with real numbers and then define complex numbers as ordered pairs of real numbers with certain algebraic properties.
I wanted to ask if there is a way to define complex numbers without reference to real numbers. What I have in mind is a set of axioms for complex numbers which would would define them as a kind of field. Like real numbers are often axiomatized as a complete ordered field. Does such axioms exist?
Please note that I do not, at this moment, posses any significant knowledge of abstract algebra or any other higher mathematics, beyond say hearing about the names of some things.