  It is shown that self-dual theories generalize to four dimensions both the
conformal and analytic aspects of two-dimensional conformal field theories. In
the harmonic space language there appear several ways to extend complex
analyticity (natural in two dimensions) to quaternionic analyticity (natural in
four dimensions). To be analytic, conformal transformations should be realized
on $CP^3$, which appears as the coset of the complexified conformal group
modulo its maximal parabolic subgroup. In this language one visualizes the
twistor correspondence of Penrose and Ward and consistently formulates the
analyticity of Fueter.
