Lucia and Xu (1992) [Recent Advances in Global Optimization. Princeton Univ. Press.] have studied the complex domain behavior of various fixed-point methods for solving the Soave-Redlich-Kwong equation of state and show that the bifurcation of a pair of roots into the complex plane coincide with a phase transition. Lucia and Taylor (1992) [European Symp. Comp. Aided Process Engng, pp. S387-S394] demonstrated that the iterative use of complex roots to cubic equations of state within flash calculations offers significant advantages in convergence in single and multiphase regions near to the critical region. In a sequel to these papers we bring new meaning to the well-used phrase "complex distillation" by carrying out Newton-based multi-stage distillation calculations almost entirely in the complex domain! This removes discontinuities inherent in real domain modes including the need for bounding mole fractions to lie between zero and one. In addition, Cauchy-Riemann conditions remove the one-sided differentiability in some models and thermophysical properties functions at phase boundaries. Numerical examples and geometric illustrations are used to illustrate significant features of complex distillation calculations.