I know that (up to isomorphism) there is only one irreducible $\mathbb{R}$-representations of $\mathbb{R}(n)$ and it is $\mathbb{R}^n.$
I know that (up to isomorphism) there is only one irreducible $\mathbb{H}$-representations of $\mathbb{H}(n)$ and that is $\mathbb{H}^n.$
I want to know how I can classify the irreducible $\mathbb{C}$-representations of $\mathbb{H}(n)$ and $\mathbb{R}(n)?$
Any help or reference would be appreciated.
Neither $\mathbb{H}(n)$ nor $\mathbb{R}(n)$ are $\mathbb{C}$-algebras. So I assume you want $\mathbb{R}$-algebra homomorphisms from these into $\mathrm{End}_{\mathbb{C}}(V)$ for complex vector spaces $V$. Since multiplication by $i$ commutes with multiplication by the algebra elements, we're really talking about complex representations of the complexifications $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}(n)$ and $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{R}(n)$, which are $\mathbb{C}$-algebras. The latter is just $\mathbb{C}(n)$, so the only irreducible representation will be the obvious one $\mathbb{C}^n$. The former is $(\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H})(n)$, and $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}\cong\mathbb{C}(2)$ so this is isomorphic to $(\mathbb{C}(2))(n)\cong\mathbb{C}(2n)$, which has exactly one irreducible representation $\mathbb{C}^{2n}$ of real dimension $4n$, which matches the $\mathbb{H}$-representation $\mathbb{H}^n$ of $\mathbb{H}(n)$ simply restricted to the scalars $\mathbb{C}\subset\mathbb{H}$.