We establish a new theory of regularity for elliptic complex valued second order equations of the form L=divA(∇⋅), when the coefficients of the matrix A satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as Lp-dissipativity. Precisely, the regularity result is a reverse Hölder condition for Lp averages of solutions on interior balls, and serves as a replacement for the De Giorgi–Nash–Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studied necessary and sufficient conditions for Lp-dissipativity of second order complex coefficient operators and systems. Recently, Carbonaro and Dragičević introduced a condition they termed p-ellipticity, and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators. Their p-ellipticity condition is exactly our strengthened version of Lp-dissipativity. The regularity results of the present paper are applied to solve Lp Dirichlet problems for L=divA(∇⋅)+B⋅∇ when A and B satisfy a Carleson measure condition, which previously was known only in the real valued case. We show solvability of the L2 Dirichlet problem, as well as solvability of the Lp Dirichlet boundary value problem for p in the range where A is p-elliptic.