  We give a pedagogical presentation of the irreducible unitary representations
of $\mathbb{C}^4\rtimes\mathbf{Spin}(4,\mathbb{C})$, that is, of the universal
cover of the complexified Poincar\'e group
$\mathbb{C}^4\rtimes\mathbf{SO}(4,\mathbb{C})$. These representations were
first investigated by Roffman in 1967. We provide a modern formulation of his
results together with some facts from the general Wigner-Mackey theory which
are relevant in this context. Moreover, we discuss different ways to realize
these representations and, in the case of a non-zero "complex mass", we give a
detailed construction of a more explicit realization. This explicit realization
parallels and extends the one used in the classical Wigner case of
$\mathbb{R}^4\rtimes\mathbf{Spin}^0(1,3)$. Our analysis is motivated by the
interest in the Euclidean formulation of Fermionic theories.
