Massless field equations are fundamental in particle physics. In
Clifford analysis, the Euclidean version of these equations has been
dealt with but it is not clear, even in dimension 4, what should be the
right analogue of massless field equations for fields with values in a
general irreducible Spin(4)-module. The main aim of the paper is to
explain that a good possibility is to take the so-called generalized
Cauchy-Riemann equations proposed a long time ago by E. Stein and G.
Weiss. For this choice of the equations, we show that their polynomial
solutions form different irreducible Spin(4)-modules. This is an
important step in developing the corrresponding function theory.