In this paper we show how hyper-complex analysis can be used to explicitly construct self-dual $SU(2)$-Yang-Mills instanton solutions on certain classes of conformally flat $4$-manifolds.
We shall use a hyper-complex argument principle to establish a natural link between the fundamental solutions of $D \Delta f = 0$ and the second Chern class of the $SU(2)$ principal bundles over these manifolds. The considered base manifolds of the bundles are not simply-connected, in general.
Actually, the paper summarizes an extension of the
corresponding results of G\"ursey and Tze on a hyper-complex
analytical description of $SU(2)$ instantons.