We consider a complex version of a Dirac-Kähler-type equation, the eight-component complex Dirac-Kähler equation with a nonvanishing mass, which can be decomposed into two Dirac equations by only a nonunitary transformation. We also write an analogue of the complex Dirac-Kähler equation in five dimensions. We show that the complex Dirac-Kähler equation is a special case of a Bhabha-type equation and prove that this equation is invariant under the algebra of purely matrix transformations of the Pauli-Gürsey type and under two different representations of the Poincaré group, the fermionic (for a two-fermion system) and bosonic \documentclass[12pt]{minimal}
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$$\mathcal{P}$$
\end{document}-representations. The complex Dirac-Kähler equation is also written in a manifestly covariant bosonic form as an equation for the system (\documentclass[12pt]{minimal}
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$$\mathcal{B}$$
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$$\mathcal{V}$$
\end{document}μ) of irreducible self-dual tensor, scalar, and vector fields. We illustrate the relation between the complex Dirac-Kähler equation and the known 16-component Dirac-Kähler equation.