'Higher order' complex numbers I recently learned that complex numbers can also be represented as matrices of the form: $$\begin{pmatrix} x & -y \\ y & x\end{pmatrix}$$ where complex multiplication corresponds to matrix multiplication and conjugation corresponds to transposition. I thought it was interesting how this representation corresponds to the $2 \times 2$ unit rotation matrix:
$$\begin{pmatrix} \cos \theta & -\sin\theta \\ \sin \theta & \cos \theta \end{pmatrix}$$
I was wondering if, then, there is a higher order version of the complex numbers? e.g.  numbers that correspond to the $3 \times 3$ unit rotation matrix:
$$\begin{pmatrix} \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta \\ \cos \theta \cos \phi & \cos \theta \sin \phi & - \sin \theta \\ -\sin \phi & \cos \phi & 0\end{pmatrix}$$ etc...

Usually with even number of dimensions you have better results than with odd. Thus, you do not have anything useful or interesting in 3 dimensions. But in 4 dimensions you have multiple interesting algebras, some of which are more or less numbers-like.
Eevee Trainer has already mentioned quaternions, which are a division algebra but not commutative, but there are also other approaches which give algebras with different properties in 4D.

*

*First of all, possibly the simplest approach is to consider the $2\times2$ real matrices themselves, which is a 4D space. Matrix operations on $2\times2$ matrices are isomorphic to so-called split-quaternions:

$\begin{align}
\boldsymbol{1} =\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad&\boldsymbol{i} =\begin{pmatrix}0&1\\-1&0\end{pmatrix},\\
\boldsymbol{j} =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad&\boldsymbol{k} =\begin{pmatrix}1&0\\0&-1\end{pmatrix}.\end{align}$
They are neither commutative, nor a division-algebra, but allow to apply any functions that can be generalized to square matrices. They have zero divisors, nilpotents and idempotents.


*Dual complex numbers. Formed by adding the dual unity $\varepsilon$ such that $\varepsilon^2=0$ to complex numbers. They can be represented by matrices of the form
$\left(
\begin{array}{cc}
 u & w \\
 0 & v \\
\end{array}
\right)$, where $u,v,w$ are complex numbers or matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 -b & a & -d & c \\
 0 & 0 & a & b \\
 0 & 0 & -b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 0 & a & 0 & b \\
 -b & -d & a & c \\
 0 & -b & 0 & a \\
\end{array}
\right)$ where $a,b,c,d$ are real numbers ($a$ is real part, $b$ is imaginary part, $c$ is dual part, $d$ is imaginary dual part). This system is commutative but has zero divisors and nilpotents.


*Tessarines. Formed by adding split-complex unity $j$ such as $j^2=1$ to complex numbers. Can be represented as matrices of the form $\left(
\begin{array}{cc}
 u & w \\
 w & u \\
\end{array}
\right)$, where $u,w$ are complex numbers or matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 -b & a & -d & c \\
 c & d & a & b \\
 -d & c & -b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 c & a & d & b \\
 -b & -d & a & c \\
 -d & -b & c & a \\
\end{array}
\right)$ where $a,b,c,d$ are real numbers ($a$ is real part, $b$ is imaginary part, $c$ is split-imaginary part, $d$ is the coefficient of $ij$ or $-ij=k$). This system is commutative algebra, but it has zero divisors and idempotents.
If one adds another complex unity $i_1$ such that $i_1^2=-1$ instead of split-complex unity $j$, one will get bicomplex numbers, which are isomorphic to this system (the split-complex unity will arise automatically as $j=-i i_1$).


*Dual split-complex numbers. They are formed by adding split-complex unity $j$ and dual unity $\varepsilon$ to real numbers. They can be represented as real matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 b & a & d & c \\
 0 & 0 & a & b \\
 0 & 0 & b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 0 & a & 0 & b \\
 b & d & a & c \\
 0 & b & 0 & a \\
\end{array}
\right)$ where $a,b,c,d$ are real numbers ($a$ is real part, $b$ is split-imaginary part, $c$ is dual part, $d$ is split-imaginary dual part).
This is a commutative algebra but with zero divisors, nilpotents and idempotents.


*4D split-complex numbers. These are formed by adding two split-complex unities $j$ and $j_1$ such that $j^2=j_1^2=1$. They are isomorphic to the matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 b & a & d & c \\
 c & d & a & b \\
 d & c & b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 c & a & d & b \\
 b & d & a & c \\
 d & b & c & a \\
\end{array}
\right).$ This is a commutative system, with zero divisors and idempotents.


*4D dual numbers. These are formed by adding two dual unities to real numbers: $\varepsilon$ and $\varepsilon_1$, such that $\varepsilon^2=\varepsilon_1^2=\varepsilon\varepsilon_1=0$. They are isomorphic to the real matrices of the following form:
$\left(
\begin{array}{cccc}
 a & b & c & d \\
 0 & a & 0 & c \\
 0 & 0 & a & b \\
 0 & 0 & 0 & a \\
\end{array}
\right)$. This system is commutative but has zero divisors and nilpotents.
