Representation theory of infinite-dimensional algebras motivates the present development of quaternionic analysis. We recall the Fueter quaternionic analogue of the Cauchy integral formula and consider its generalizations. Our study extensively uses representation theory of the conformal group of quaternions. In particular, intertwining operators for tensor products of certain representations of the conformal group allow us to define quaternionic algebras of functions. Quaternionic dilogarithm, box Feynman diagram, and other relations to four-dimensional conformal field theory in physics appear naturally in our development of quaternionic analysis.