The power flow problem can be compactly formulated using complex variables; its solution is however commonly computed using the real variable Newton-Raphson algorithm, where the complex variables are replaced by their real valued rectangular or polar coordinates. A Newton-Raphson solution in complex variables is hindered by the fact that the power flow equations are non-analytic in their phasor voltages, i.e., they do not have a Taylor series expansion in terms of these complex variables alone. By using Wirtinger calculus, the power flow equations can be expanded in terms of the phasor voltages and their conjugates, thus allowing the implementation of the complex variable Newton-Raphson algorithm. This paper specializes the solution of the complex variable Newton-Raphson power flow to distribution networks, and includes the particularities of modeling line drop compensators, reactive power limited generation, and voltage dependent loads. Numerical results presented on distribution networks with up to 3139 nodes show that the complex implementation is better suited to modern processors that employ single instruction multiple data (SIMD) processor extensions; in particular SIMD instructions result in a complex variable formulation that is faster than the classical real variable formulation, yet it has the distinct advantage of a simple software implementation.