  We show that 4--dimensional conformal field theory is most naturally
formulated on Kulkarni 4--folds, i. e. real 4--folds endowed with an integrable
quaternionic structure. This leads to a formalism that parallels very closely
that of 2--dimensional conformal field theory on Riemann surfaces. In this
framework, the notion of Fueter analyticity, the quaternionic analogue of
complex analyticity, plays an essential role. Conformal fields appear as
sections of appropriate either harmonic real or Fueter holomorphic quaternionic
line bundles. In the free case, the field equations are statements of either
harmonicity or Fueter holomorphicity of the relevant conformal fields. We
obtain compact quaternionic expressions of such basic objects as the
energy-momentum tensor and the gauge currents for some basic models in terms of
Kulkarni geometry. We also find a concise expression of the conformal anomaly
and a quaternionic 4--dimensional analogue of the Schwarzian derivative
describing the covariance of the quantum energy-momentum tensor. Finally, we
analyse the operator product expansions of free fields.
