A real analogue of a $\mathbf C^*$-algebra is a real Banach algebra $A$, together with an $\mathbf R$-linear anti-involution $*:A\to A$ such that for all $0\neq a\in A$, $\|a^*a\|>0$. For example, Hamilton's quaternions with the usual anti-involution.
Such an algebra can be used to construct a $\mathbf C^*$-algebra in the usual sense; take $A\otimes_{\mathbf R}\mathbf C$ and extend $*$ to be conjugate linear using $(a\otimes z)^*=a^*\otimes \bar z$. This algebra could be called the complexification of $A$.
I am looking for a reference which discusses the Wedderburn decomposition and module theory of finite dimensional algebras of this kind. I would prefer a reference where the relationship between the module theory of the algebra and its complexification are also discussed.
What I am really interested in is understanding how far the theory of the Frobenius-Schur index (which is for group algebras) can be developed in this setting.