In this paper, an exact expression for the first probability density function of the solution stochastic process to a randomized homogeneous linear second-order complex differential equation is determined. To complete the probabilistic analysis, the first probability density functions of the real and complex contributions of the solution stochastic process are also calculated. To compute the densities, the random variable transformation method is applied under general hypothesis, all coefficients and initial conditions are absolutely continuous complex random variables. The capability of the theoretical results established is demonstrated by several numerical examples. Finally, we show the applicability of the method in engineering, by analysing the solution of a randomized simple harmonic oscillator, defined on the complex domain, and comparing our results with those obtained by using Monte Carlo simulations.