  Considering two-dimensional electron gases under a perpendicular magnetic
field, we pinpoint a specific kind of long-range bipartite entanglement of the
electronic motions. This entanglement is achieved through the introduction of
bicomplex spinorial eigenfunctions admitting a polar decomposition in terms of
a real modulus and three real phases. Within this bicomplex geometry the
cyclotron motions of two electrons are intrinsically tied, so that the
highlighted eigenstates of the kinetic energy operator actually describe the
free motion of a genuine electron pair. Most remarkably, these states embody
phase singularities in the four-dimensional (4D) space, with singular points
corresponding to the simultaneous undetermination of the three phases. Because
the entanglement between the two electrons forming a pair, as well as the
winding and parity quantum numbers characterizing the 4D phase singularity, are
topological in nature, we expect them to manifest some robustness in the
presence of a smooth disorder potential and an electron-electron interaction
potential. The relevance of this effective approach in terms of 4D vortices of
electron pairs is discussed in the context of the fractional quantum Hall
effect.
