  Using the Cayley-Dickson construction we rephrase and review the
non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I.
Zahed, Nucl.Phys. B $\textbf{501}$, 603 (1997)], that generalizes the free
probability calculus to asymptotically large non-hermitian random matrices. The
main object in this generalization is a quaternionic extension of the R
transform which is a generating function for planar (non-crossing) cumulants.
We demonstrate that the quaternionic R transform generates all connected
averages of all distinct powers of $X$ and its hermitian conjugate $X^\dagger$:
$\langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots
\rangle\rangle$ for $N\rightarrow \infty$. We show that the R transform for
gaussian elliptic laws is given by a simple linear quaternionic map
$\mathcal{R}(z+wj) = x + \sigma^2 \left(\mu e^{2i\phi} z + w j\right)$ where
$(z,w)$ is the Cayley-Dickson pair of complex numbers forming a quaternion
$q=(z,w)\equiv z+ wj$. This map has five real parameters $\Re e x$, $\Im m x$,
$\phi$, $\sigma$ and $\mu$. We use the R transform to calculate the limiting
eigenvalue densities of several products of gaussian random matrices.
