We give a possible extension of the definition of quaternionic power
series, partial derivatives and vector fields in the case of two (and then several)
non commutative (quaternionic) variables. In this setting we also investigate
the problem of describing zero functions which are not null functions in the for-
mal sense. A connection between an analytic condition and a graph theoretic
property of a subgraph of a Hamming graph is shown, namely the condition
that polynomial vector field has formal divergence 0 is equivalent to connect-
edness of subgraphs of Hamming graphs H(d, 2). We prove that monomials in
variables z and w are always linearly independent as functions only in bidegrees
(p, 0), (p, 1), (0, q), (1, q) and (2, 2).
