  Holomorphic quaternion functions only admit affine functions; thus, the
M\"obius transformation for these functions, which we call quaternionic
holomorphic transformation (QHT), only comprises similarity transformations. We
determine a general group $\mathsf{X}$ which has the group $\mathsf{G}$ of QHT
as a particular case. Furthermore, we observe that the M\"obius group and the
Heisenberg group may be obtained by making $\mathsf{X}$ more symmetric. We
provide matrix representations for the group $\mathsf{X}$ and for its algebra
$\mathfrak{x}$. The Lie algebra is neither simple nor semi-simple, and so it is
not classified among the classical Lie algebras. They prove that the group
$\mathsf{G}$ comprises $\mathsf{SU}(2,\mathbb{C})$ rotations, dilations and
translations. The only fixed point of the QHT is located at infinity, and the
QHT does not admit a cross-ratio. Physical applications are addressed at the
conclusion.
