It is shown that the Dirac operator becomes 2 × 2 matrix acting on the space of chiral fermions in the previously-proposed quaternionic formulation of the Dirac theory. The interaction part of thus-obtained Dirac operator formally corresponds to the 2 × 2 matrix of the generalized one-form of Coquereaux et al. in the non-commutative geometric approach to the electro-weak unification. It is pointed out that the mathematical formulation of Coquereaux et al. allows to introduce an arbitrary parameter which can be reduced to unity unless it vanishes and whose presence spoils a unique construction of Yang-Mills-Higgs theory in their approach. For instance, one may choose the parameter to vanish, resulting in only pure Yang-Mills theory