The concept of the Q field is introduced as a 2×2 matrix representation of the variable basis of vectors satisfying the rule of multiplication of quaternion imaginary numbers and as an element of the group of transformations of the basis preserving the invariance of this multiplication rule. The rule for projecting such matrices on a given direction is determined with the help of the characteristic functions of the matrices-vectors of the Q field. The differential structure of Q fields is studied. The theory developed is illustrated by an example of a model-topological classification of particles according to the magnitude of their spin. © 1986 Plenum Publishing Corporation.