We present a scheme of biquaternionic algebrodymamics based on a nonlinear generalization of the Cauchy-Riemann "holomorphy" conditions considered therein as fundamental field equations. The automorphism group SO(3,C) of the biquaternion algebra acts as a proper Lorentz group on a real space whose coordinates are bilinear in the complex coordinates of biquaternionic vector space. A new invariant of Lorentz transformations then arises - the geometric phase. This invariant can be responsible for the quantum properties of particles associated in this approach with field singularities. Some new notions are introduced, related to "hidden" complex dynamics: "observable" space-time, the ensemble of identical correlated particles-singularities ("duplicons") and others.