Let $R=k[[u,v]]$ be a power series ring over algebraically closed field of characteristic zero. The quaternionic $R$-algebra is $A=R\langle x,y\rangle/I$, where $I=(x^2-a, y^2-b, xy+yx-2c)$ and $a,b,c\in R$. I am interested in existence of such algebras for general $a,b,c$ and an explicit matrix representation for them. It is possible to construct some specific examples as the rings of invariants (see for example here, page 74) but what is the situation in general?