There is a characterization of the real number system：complete ordered field. And the complete ordered field is unique up to isomorphism. I'm trying to charaterise the complex numbers in a similar way.
In this paper, the author introduces the complex number system as a field $\mathbb{F}$ together with an automorphism $T:\mathbb{F}\rightarrow\mathbb{F}$ with certain properties ($T$ is intended to act as the complex conjugate operator) and a linear ordering on the subset $\{z\in\mathbb{F}:T(z)=z\}$ that turns it into a complete ordered field.
This method makes sense to me. However, I find it difficult to see why the introduction of the conjugate operator is a crucial step in defining a complex number system. And I came up with another method that seems plausible to me and I hope to receive some suggestions.
Definition: A complex number system $(C,+,\cdot,\lt)$ is a struture with the following postulates:
a)$(C,+,\cdot)$ is a field
b)there is a subset $R$ of $C$ such that $(R,+|_{R\times R},\cdot|_{R\times R},\lt)$ is a complete ordered field
c)there is an element $i\in C$ such that $i^2=-1$ ($1$ is the multiplicative identity)
d)for every $z\in C$, there is a unique $a\in R$ and a unique $b\in R$ such that $z=a+bi$ ($a$ is called the real part of $z$, denoted $\mathcal{Re(z)}$, and $b$ is called the imaginary part of $z$, denoted $\mathcal{Im(z)}$)
e)for every $z\in C$, $z\in R$ iff $\mathcal{Im(z)}=0$