  It is known that quaternions represent rotations in 3D Euclidean and
Minkowski spaces. However, product by a quaternion gives rotation in two
independent planes at once and to obtain single-plane rotations one has to
apply by half-angle quaternions twice from the left and on the right (with its
inverse). This 'double cover' property is potential problem in geometrical
application of split quaternions, since (2+2)-signature of their norms should
not be changed for each product. If split quaternions form proper algebraic
structure for microphysics, representation of boosts in (2+1)-space leads to
the interpretation of the scalar part of quaternions as wavelength of
particles. Invariance of space-time intervals and some quantum behavior, like
noncommutativity and fundamental spinor representation, probably also are
algebraic properties. In our approach the Dirac equation represents the
Cauchy-Riemann analyticity condition and the two fundamental physical
parameters (speed of light and Planck's constant) appear from the requirement
of positive definiteness of quaternionic norms.
