  The standard momentum operator $-i \nabla$ can not be accepted as observable
in relativistic quantum mechanics of the Majorana particle. Instead, one can
use axial momentum operator recently proposed in Phys. Lett. A {\bf383}, 1242
(2019). In the present paper we report several new results related to the axial
momentum which elucidate its usability. First, a new motivation for the axial
momentum is given, and the Heisenberg uncertainty relation checked. Next, we
show that the general solution of time evolution equation in the axial momentum
basis has a connection with quaternions. Single traveling plane waves are not
possible in the massive case, but there exist solutions which consist of
asymmetric pair of plane waves traveling in opposite directions. Finally,
pertinent real orthogonal and irreducible representation of the Poincar\'e
group -- consistent with the lack of antiparticle -- is unveiled.
