The possibility of a new quantum mechanical description in which the observables are realized as self-adjoint operators in quaternionic Hilbert spaces is investigated from the point of view of relativistic invariance. The problem is first formulated in the natural language of quantum mechanics, i.e. the one of proposition systems. This is followed by the theory of projective group representation and their reduction to unitary group representations in quaternionic Hilbert spaces. The general theory is then applied to the group of special relativity which leads to the classification of elementary systems in this theory. This last step will be carried out in a further paper to appear in this journal.