About the construction of $\mathbb{C}$ by $2\times 2$ real matrices in "Linear Algebra" by Ichiro Satake.

There is a (naive philosophical) analogy with object-oriented programming that I didn't see in linked posts and which may be worth noting for posterity. In object-oriented programming, a class has a public interface specifying the operations performable on or with objects of that type, and a private interface giving an implementation of the public interface in terms of other "general" data and instructions.

Here, for example, the public interface of a complex number would include addition, multiplication, taking real and imaginary parts, and detecting equality. (The public interface might also include auxiliary and subsidiary operations, such as forming the complex conjugate, computing the magnitude, finding the principal branch of argument, etc.)

Specifying a public interface for complex numbers corresponds mathematically to specifying axioms that characterize the complex field $(\mathbf{C}, +, \cdot)$. At risk of mis-using an established philosophical term, an "operationalist" perspective might be that any mathematical structure that fulfills the promise of the public interface is a complex number.

The private interface of a complex number, by contrast, would be a construction (in set theory, for example) of entities and operations that implement the public interface.

Here we come to the question: Satake has mentioned two different implementations of the "complex number class":

Certain $2 \times 2$ real matrices, equipped with addition and multiplication of matrices, functions returning the entries in the first column, and equality of matrices;
Formal expressions $(a_{1}, a_{2}) = a_{1} + a_{2}i$ with $a_{1}$ and $a_{2}$ real, equipped with vector addition, a multiplication rule formally obeying the associative, commutative, and distributive laws for real numbers and the identity $i^{2} = -1$, functions returning Cartesian coordinates, and equality of ordered pairs.

Again at risk of mis-using an established term, an "essentialist perspective" might view these implementations as distinct; the underlying entities are "apples and oranges." To write $A = \alpha$ is to assert that some real $2 \times 2$ matrix is an ordered pair of real numbers.

How do we know these implementations are abstractly equivalent? Satake has a particularly elegant perspective: Each implementation is itself built as a set of formal linear combinations $a_{1}I + a_{2}J$ of two entities, one ($I$) acting as a multiplicative identity, the other ($J$) a square root of the additive inverse of $I$. If we define addition, multiplication, real and imaginary parts, and equality of these entities in a uniform formal way, we implement the public interface of the complex field.

Coda: What forced us to pick $$ J = \left[\begin{array}{@{}rr@{}} 0 & -1 \\ 1 & 0 \\ \end{array}\right]? $$ Nothing except simplicity at the level of implementation. It's easily checked that if $P$ is an invertible $2 \times 2$ real matrix, then $P^{-1}JP$ also serves as a square root if $-I$, and conversely. (If it matters, there is a four-parameter family of invertible matrices, and of these a two-parameter family commute with $J$, so there is a two-parameter family of choices of square root of $-I$.) If we like, we may pick any matrix $X$ satisfying $X^{2} = -I$ and define a complex number to be a formal linear combination $a_{1}I + a_{2}X$.

The final point, where nitpicking now matters, concerns equality: As "abstract" complex numbers, $a_{1}I + a_{2}J$ and $a_{1}I + a_{2}X$ are equal. In more detail, under the bijection $a_{1}I + a_{2}J \leftrightarrow a_{1}I + a_{2}X$, any true "complex" statement for one implementation yields a true statement in the other implementation. As $2 \times 2$ real matrices, on the other hand, these entities are probably not equal. Equality is not a universal concept, but a binary relation defined in some context.

You should understand it as motivational remark why the author chose the matrix approach to introduce the complex numbers.

It is clear that the matrix approach produces a subfield $\mathbf C \subset M(\mathbb R,2)$ of the ring of real $2\times 2$-matrices. He gets multiplication for free, it is just matrix multiplication. And in fact $\mathbf C$ is canonically isomorphic to the "standard" $\mathbb C$ having as elements all $x+iy$ with $x,y \in \mathbb R$.

But is it a straightforward idea to consider matrices of the form $\begin{pmatrix}a_1&-a_2\\a_2&a_1\end{pmatrix}$ to introduce the complex numbers? This may be a philosophical question. The author tries to motivate it as follows.

Of course, as real vector spaces we can identify $\mathbb C$ and $\mathbb R^2$ via $\xi = x_1 + x_2i \mapsto \begin{pmatrix}x_1 \\x_2\end{pmatrix}$. Complex multiplication by $\alpha = a_1 + a_2i$ is a $\mathbb C$-linear map $L_\alpha : \mathbb C \to \mathbb C, L_\alpha(\xi) = \alpha \xi$, which can be regarded as an $\mathbb R$-linear map $L_\alpha : \mathbb R^2 \to \mathbb R^2$. Its matrix represenation in $M(\mathbb R, 2)$ is $A_\alpha = \begin{pmatrix}a_1&-a_2\\a_2&a_1\end{pmatrix}$. Thus we get an injection $$\phi : \mathbb C \to M(\mathbb R, 2), \phi(\alpha) = A_\alpha .$$ It is obvious that $\phi(\alpha + \beta) = \phi(\alpha) + \phi(\beta)$. Moreover, $L_{\alpha\beta}(\xi) = \alpha \beta \xi = \alpha L_\beta(\xi) = L_\alpha L_\beta(\xi)$. In other words $\phi(\alpha \beta) = \phi(\alpha) \phi( \beta)$ in $M(\mathbb R, 2)$ which shows that $\phi$ is a ring homomorphism. Its image $\mathbf C = \phi(\mathbb C)$ is clearly isomorphic to $\mathbb C$.