Is commutativity of observables the main feature, which separate classical mechanics from quantum?

In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Diracs proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit ϵ such that ϵ2 = 0. Thus, the crucial role in quantum theory shall be attributed to the usage of complex numbers.