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.
Categories: math-ph math.MP