Algebraic quaternion construction

First, note if $A$ is any unital associative algebra and $a\in A^{\times}$ a unit, then conjugation ($x\mapsto axa^{-1}$) is an algebra automorphism - we check this the same way we do for groups (but also check linearity).

The Clifford algebra $\mathcal{C}\ell(n)$ is the free associative algebra generated by $n$ anticommuting square roots of negative one. That is, it may be defined as the quotient of a free associative algebra:

$$ \mathbb{R}\langle x_1,\cdots,x_n\rangle/(x_i^2+1,x_ix_j+x_jx_i) $$

so that the relations $x_i^2=-1$ and $x_jx_i=-x_ix_j$ (when $i\ne j$) are imposed. Note the free associative algebra generated by $\{x_1,\cdots,x_n\}$ is basically the "noncommutative" polynomial ring in them, although of course they commute if there's only one variable.

Then $\mathcal{C}\ell(0)=\mathbb{R}$ and $\mathcal{C}\ell(1)=\mathbb{C}$ (generated by $i$) and $\mathcal{C}\ell(2)=\mathbb{H}$ (generated by $i,j$).

Since $Q_8\subset\mathbb{H}^{\times}$ is generated by $i,j$ as well, it spans $\mathbb{H}$, so we can define $\mathbb{H}$ by a quotient of the group algebra $\mathbb{R}[Q_8]$, however this is about as natural as defining $\mathbb{C}$ as a quotient of the group algebra $\mathbb{R}[\{\pm1,\pm i\}]$.

They Cayley-Dickson construction proceeds to build algebras $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}\subset\mathbb{S}$ where the octonions $\mathbb{O}$ lose full associativity (although they retain "alternativity," i.e. every subalgebra generated by $2$ elements is associative), and sedenions $\mathbb{S}$ begin introducing zero divisors. The octonions are $8$-dimensional and the sedenions are $16$-dimensional unital algebras.

Clifford algebras also begin introducing zero divisors; $\mathcal{C}\ell(3)$ is $\cong\mathbb{H}\oplus\mathbb{H}$ (good exercise). There is an interesting "mod $8$" periodicity to them (not up to isomorphism, but up to Morita equivalence) which may be described by what Baez calls the "Clifford clock".

You can also do Clifford algebras over other fields, or describe them in a "coodinate-free" way as a quotient of a tensor algebra over a vector space with a quadratic form, you can describe how they combine as super-algebras with super-tensor products, etc.