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][1], page 74) but what is the situation in general?

  [1]: http://web.maths.unsw.edu.au/~danielch/thesis/kenneth.pdf