Phenomenology from Dirac equation with Euclidean-Minkowskian "gravity phase"

Over the past decades, many authors advertised models on complexified spacetime algebras for use in describing gravity. This work aims at providing phenomenological support to such claims, by introducing a one-parameter real phase $\alpha$ to the conventional Dirac equation with $\frac{1}{r}$-type potential. This phase allows to transition between Euclidean ($\alpha=0,\pm\pi,\pm2\pi,\ldots$) and Minkowskian ($\alpha=\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\ldots$) geometry, as two distinct cases that one may expect from some complexified spacetime. The configuration space is modeled on $4\times4$ matrix algebra over the bicomplex numbers, $\mathbb{C}\oplus\mathbb{C}$. Spin-$\frac{1}{2}$ Coulomb scattering (Rutherford scattering) in Born approximation is then executed. All calculations are done ``from scratch'', as they could have been done some 85 years ago. By removing elegance from field theory that has since become customary, this paper aims at remaining as generally applicable as possible, for a wide range of candidate models that contain such a phase $\alpha$ in one way or another. Results for backscattering and cross section at high energies are compared with results from General Relativity calculations. Effects on intergalactic gas distribution and momentum transfer from scattering high-energy leptons are sketched.