A one-parametric stochastic regularized dynamics of the interface in the Hele-Shaw cell is introduced. The short-distance regularization suggested by the aggregation model stabilizes the growth by preventing the formation of cusps at the interface and makes the interface dynamics chaotic. The introduced stochastic growth process generates universal complex patterns with the well-developed fjords of oil separating the fingers of water. In a long time asymptotic, by coupling a conformal field theory to the stochastic growth process, we introduce a set of observables (the martingales), whose expectation values are constant in time. The martingales are closely connected to degenerate representations of the Virasoro algebra and can be written in terms of conformal correlation functions. A direct link between Laplacian growth and conformal Liouville field theory with the central charge c≥25 is proposed.