I am currently researching the vector space $\mathbb{C}^{3}$ and I was wondering if it is possible to generate a scheme of unifying the rigid transformations in $\mathbb{C}^{3}$. I know that in the extended complex plane one can perform Möbius transformations to produce rotations, translations, and reflections, but can this also be done in an elegant way for bivectors?
My question, then, is this: if $\textbf{e}_{x}$, $\textbf{e}_{y}$, and $\textbf{e}_{z}$ were the respective planes of $\mathbb{C}^{3}$, would it be possible to then perform transformations on the x, y, z planes together, independently, and in any arbitrary order using a unified algebraic scheme? A friend of mine mentioned exterior algebras for this, but I am unfamiliar with the process. Can one mix Möbius transformations and rotation matrices to perform this, and if so, can I be pointed in the right direction with examples in the literature?
Thank you!