  Consider an iterated function system consisting of similarities on the
complex plane of the form $g_{i}(z) = \lambda_i z + t_i,\ \lambda_i, t_i \in
\mathbb{C},\ |\lambda_i|<1, i=1,\ldots, k$. We prove that for almost every
choice of $(\lambda_1, \ldots, \lambda_k)$ 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.
