According to the Wedderburn–Artin Theorem:

Any finite-dimensional simple algebra is isomorphic to the matrix algebra $M_n(D)$ for some division ring $D$.

So it is enough to look for examples in this setting. In the particular case when $D=K$ is the base field, it is a central-simple algebra. If such field $K$ contains a primitive $n$-th root of 1 (let us denote it by $\varepsilon$) one can define the following matrices:

$$A=\begin{pmatrix} \varepsilon^{n-1} & 0 & \cdots & 0 & 0\\ 0& \varepsilon^{n-2} & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & \varepsilon^{n-1} & 0\\ 0 & 0 & \cdots & 0 & 1\\ \end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\\ 1 & 0 & 0 & \cdots & 0\\ % \end{pmatrix}.$$

The above matrices satisfy $A^n=B^n=I_n$ and $AB=\varepsilon BA$.

In particular, the set $\{A^kB^l\,|\,0\leq k, l \leq n-1\}$ is a basis of $M_n(K)$.

The above basis elements satisfy $A^kB^l A^rB^s=\varepsilon^{-lr}A^{k+r}B^{l+s}$.

This may be seen as a generalization of such properties of basis elements of quaternions mentioned in the question.

Remark: this algebra is isomorphic to the cyclic algebras mentioned in @abx's answer. The isomorphism can be defined by sending $A$ to $y$ and $B$ to $x$. Nevertheless, it seems more natural to me to consider an example on matrix algebras, in light of the Wedderburn-Artin Theorem.

According to the Wedderburn–Artin Theorem:

Any finite-dimensional simple algebra is isomorphic to the matrix algebra $M_n(D)$ for some division ring $D$.

So it is enough to look for examples in this setting. In the particular case when $D=K$ is the base field, it is a central-simple algebra. If such field $K$ contains a primitive $n$-th root of 1 (let us denote it by $\varepsilon$) one can define the following matrices:

$$A=\begin{pmatrix} \varepsilon^{n-1} & 0 & \cdots & 0 & 0\\ 0& \varepsilon^{n-2} & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & \varepsilon^{n-1} & 0\\ 0 & 0 & \cdots & 0 & 1\\ \end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\\ 1 & 0 & 0 & \cdots & 0\\ % \end{pmatrix}.$$

The above matrices satisfy $A^n=B^n=I_n$ and $AB=\varepsilon BA$.

By straightforward computations and using the Vandermonde determinant one can prove that the set $\{A^kB^l\,|\,0\leq k, l \leq n-1\}$ is a linear basis of $M_n(K)$.

The above basis elements satisfy $A^kB^l A^rB^s=\varepsilon^{-lr}A^{k+r}B^{l+s}$.

This may be seen as a generalization of such properties of basis elements of quaternions mentioned in the question.

Remark: this algebra is isomorphic to the cyclic algebras mentioned in @abx's answer. The isomorphism can be defined by sending $A$ to $y$ and $B$ to $x$. Nevertheless, it seems more natural to me to consider an example on matrix algebras, in light of the Wedderburn-Artin Theorem.

According to the Wedderburn-Artin Theorem:

Any finite-dimensional simple algebra is isomorphic to the matrix algebra $M_n(D)$ for some division ring $D$.

So it is enough to look for examples in this setting. In the particular case when $D=K$ is the base field, it is a central-simple algebra. If such field $K$ contains a primitive $n$-th root of 1 (let us denote it by $\varepsilon$) one can define the following matrices:

$A=\begin{pmatrix} \varepsilon^{n-1} & 0 & \cdots & 0 & 0\\ 0& \varepsilon^{n-2} & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & \varepsilon^{n-1} & 0\\ 0 & 0 & \cdots & 0 & 1\\ \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\\ 1 & 0 & 0 & \cdots & 0\\ % \end{pmatrix}$

The above matrices satisfy $A^n=B^n=I_n$ and $AB=\varepsilon BA$

In particular, the set $\{A^kB^l\,|\,0\leq k, l \leq n-1\}$ is a basis of $M_n(K)$.

The above basis elements satisfy $A^kB^l A^rB^s=\varepsilon^{-lr}A^{k+r}B^{l+s}$.

This may be seen as a generalization of such properties of basis elements of Quaternions mentioned in the question.

Remark: this algebra is isomorphic to the cyclic algebras mentioned in @abx answer. The isomorphism can be defined by sending $A$ to $y$ and $B$ to $x$. Nevertheless, it seems more natural to me to consider an example on matrix algebras, in light of the Wedderburn-Artin Theorem.

According to the Wedderburn–Artin Theorem:

Any finite-dimensional simple algebra is isomorphic to the matrix algebra $M_n(D)$ for some division ring $D$.

So it is enough to look for examples in this setting. In the particular case when $D=K$ is the base field, it is a central-simple algebra. If such field $K$ contains a primitive $n$-th root of 1 (let us denote it by $\varepsilon$) one can define the following matrices:

$$A=\begin{pmatrix} \varepsilon^{n-1} & 0 & \cdots & 0 & 0\\ 0& \varepsilon^{n-2} & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & \varepsilon^{n-1} & 0\\ 0 & 0 & \cdots & 0 & 1\\ \end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\\ 1 & 0 & 0 & \cdots & 0\\ % \end{pmatrix}.$$

The above matrices satisfy $A^n=B^n=I_n$ and $AB=\varepsilon BA$.

In particular, the set $\{A^kB^l\,|\,0\leq k, l \leq n-1\}$ is a basis of $M_n(K)$.

The above basis elements satisfy $A^kB^l A^rB^s=\varepsilon^{-lr}A^{k+r}B^{l+s}$.

This may be seen as a generalization of such properties of basis elements of quaternions mentioned in the question.

Remark: this algebra is isomorphic to the cyclic algebras mentioned in @abx's answer. The isomorphism can be defined by sending $A$ to $y$ and $B$ to $x$. Nevertheless, it seems more natural to me to consider an example on matrix algebras, in light of the Wedderburn-Artin Theorem.