  We develop quaternionic analysis using as a guiding principle representation
theory of various real forms of the conformal group. We first review the
Cauchy-Fueter and Poisson formulas and explain their representation theoretic
meaning. The requirement of unitarity of representations leads us to the
extensions of these formulas in the Minkowski space, which can be viewed as
another real form of quaternions. Representation theory also suggests a
quaternionic version of the Cauchy formula for the second order pole.
Remarkably, the derivative appearing in the complex case is replaced by the
Maxwell equations in the quaternionic counterpart. We also uncover the
connection between quaternionic analysis and various structures in quantum
mechanics and quantum field theory, such as the spectrum of the hydrogen atom,
polarization of vacuum, one-loop Feynman integrals. We also make some further
conjectures. The main goal of this and our subsequent paper is to revive
quaternionic analysis and to show profound relations between quaternionic
analysis, representation theory and four-dimensional physics.
