Consider an iterated function system consisting of similarities on the complex plane of the form $g_{i}(z) = λ_i z + t_i,\ λ_i, t_i \in \mathbb{C},\ |λ_i|<1, i=1,\ldots, k$. We prove that for almost every choice of $(λ_1, \ldots, λ_k)$ in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). As the main steps of the proof, we obtain results on the dimension and power Fourier decay of random self-similar measures on the plane, which may be of independent interest.