I find that the justification of complex field in Ahlfors’ book is quite enlightening, as it explains perfectly why we need complex field $\mathbb{C}$ and why it is THE field we need.
Justification
So far the regular approach to complex numbers has been completely uncritical as we have not questioned the existence of a number system in which the equation $x^2 + 1 = 0$ has a solution while all the rules of arithmetic remain in force.
We begin by recalling the characteristic properties of the real-number system which we denote by $\mathbb{R}$. In the first place, $\mathbb{R}$ is a field. This means that addition and multiplication are defined, satisfying the associative, commutative, and distributive laws. The numbers $0$ and $1$ are neutral elements under addition and multiplication, respectively: $\alpha+0=\alpha$, $\alpha\cdot1=\alpha$ for all $\alpha$. Moreover, the equation of subtraction $\beta+x=\alpha$ has always a solution, and the equation of division $\beta x=\alpha$ has a solution whenever $\beta\neq0$.
One shows by elementary reasoning that the neutral elements and the results of subtraction and division are unique. Also, every field is an integral domain.
These properties are common to all fields. In addition, the field $\mathbb{R}$ has an order relation $\alpha<\beta$ (or $\alpha>\beta$). It is most easily defined in terms of the set $\mathbb{R}_+$ of positive real numbers: $\alpha<\beta$ if and only if $\beta-\alpha\in\mathbb{R}_+$. The set $\mathbb{R}_+$ is characterized by the following properties: (1) $0$ is not a positive number; (2) if $\alpha\neq0$ either $\alpha$ or $-\alpha$ is positive; (3) the sum and the product of two positive numbers are positive. From these conditions one derives all the usual rules for manipulation of inequalities. In particular one finds that every square $\alpha^2$ is either positive or zero; therefore $1=1^2$ is a positive number.
By virtue of the order relation the sums $1$, $1 +1$, $1 +1 +1$, are all different. Hence $\mathbb{R}$ contains the natural numbers, and since it is a field it must contain the subfield formed by all rational numbers. Finally, $\mathbb{R}$ satisfies the following completeness condition: every increasing and bounded sequence of real numbers has a limit.
Our discussion of the real-number system is incomplete inasmuch as we have not proved the existence and uniqueness (up to isomorphisms) of a system $\mathbb{R}$ with the postulated properties. We just skip the proof here.
The equation $x^2 + 1 = 0$ has no solution in $\mathbb{R}$, for $\alpha^2 + 1$ is always positive. Suppose now that a field $\mathbb{F}$ can be found which contains $\mathbb{R}$ as a subfield, and in which the equation $x^2 + 1 = 0$ can be solved. Denote a solution by $i$. Then $x^2 + 1 = (x + i)(x - i)$, and the equation $x^2 + 1 = 0$ has exactly two roots in $\mathbb{F}$, $i$ and $-i$. Let $\mathbb{C}$ be the subset of $\mathbb{F}$ consisting of all elements which can be expressed in the form $\alpha + i\beta$ with real $\alpha$ and $\beta$. This representation is unique, for $\alpha+ i\beta = \alpha’ + i\beta’$ implies $\alpha - \alpha’ = -i(\beta - \beta’)$; hence $(\alpha - \alpha’)^2 = -(\beta - \beta’)^2$, and this is possible only if $\alpha = \alpha’, \beta = \beta’$.
The subset $\mathbb{C}$ is a subfield of $\mathbb{F}$. What is more, the structure of $\mathbb{C}$ is independent of $\mathbb{F}$. For if $\mathbb{F}’$ is another field containing $\mathbb{R}$ and a root $i’$ of the equation $x^2 + 1 = 0$, the corresponding subset $\mathbb{C}’$ is formed by all elements $\alpha+i’\beta$. There is a one-to-one correspondence between $\mathbb{C}$ and $\mathbb{C}’$ which associates $\alpha+i\beta$ and $\alpha+i’\beta$, and this correspondence is evidently a field isomorphism. It is thus demonstrated that $\mathbb{C}$ and $\mathbb{C}’$ are isomorphic.
We now define the field of complex numbers to be the subfield $\mathbb{C}$ of an arbitrarily given $\mathbb{F}$. We have just seen that the choice of $\mathbb{F}$ makes no difference, but we have not yet shown that there exists a field $\mathbb{F}$ with the required properties. In order to give our definition a meaning it remains to exhibit a field $\mathbb{F}$ which contains $\mathbb{R}$ (or a subfield isomorphic with $\mathbb{R}$) and in which the equation $x^2 + 1 = 0$ has a root.
There are many ways in which such a field can be constructed. The simplest and most direct method is the following: Consider all expressions of the form $\alpha+i\beta$ where $\alpha$, $\beta$ are real numbers while the signs $+$ and $i$ are pure symbols ($+$ does not indicate addition, and $i$ is not an element of a field). These expressions are elements of a field $\mathbb{F}$ in which addition and multiplication are defined by (1) and (2) (observe the two different meanings of the sign $+$). The elements of the particular form $\alpha+i0$ are seen to constitute a subfield isomorphic to $\mathbb{R}$, and the element $0+i1$ satisfies the equation $x^2+1=0$; we obtain in fact $(0+i1)^2=-(1 +i0)$. The field $\mathbb{F}$ has thus the required properties; moreover, it is identical with the corresponding subfield $\mathbb{C}$, for we can write
The existence of the complex-number field is now proved, and we can go back to the simpler notation $\alpha+i\beta$ where the $+$ indicates addition in $\mathbb{C}$ and $i$ is a root of the equation $x^2 +1=0$.
