  Due to the existence of incompatible observables, the propositional calculus
of a quantum system does not form a Boolean algebra but an orthomodular
lattice. Such lattice can be realised as a lattice of subspaces on a real,
complex or quaternionic Hilbert space, which motivated the formulation of real
and quaternionic quantum mechanics in addition to the usual complex
formulation. It was argued that any real quantum system admits a complex
structure that turns it into a complex quantum system and hence real quantum
mechanics was soon discarded. Several authors however developed a quaternionic
version of quantum mechanics and this version did not seem to be equivalent its
standard formulation on a complex Hilbert space.
  Motivated by some recently developed techniques from quaternionic operator
theory, we conjecture in this article that this not correct and that any
quaternionic quantum system is actually simply the quaternionification of a
complex quantum system. We then show that this conjecture holds true for
quaternionic relativistic elementary systems by applying some recent arguments
that were used to show the equivalence of real and complex elementary
relativistic systems. Finally, we conclude by discussing how the misconception
that complex and quaternionic quantum mechanics are inequivalent arose from
assuming the existence of a left multiplication on the Hilbert space, which is
physically not justified.
