Dynamics of thermal plasma jets is characterized by significant turbulent phenomena influencing various technological applications. In two types of experiments, we have analysed (i) tomographical reconstructions of plasma jet radiation captured from four sides by photodiode arrays in two planes perpendicular to the jet axis and (ii) recordings of the jet radiation acquired by a high-speed CCD camera. The analysis was based on the method of generalized correlation dimension stemming from the correlation dimension estimates by the Grassberger-Procaccia algorithm common in the domain of nonlinear dynamic systems. Our results map spatial distributions of complexity within recorded data including their temporal evolution and provide more information about the jet dynamics than, e.g., calculations of standard deviations. The most stable region of the plasma jet was identified at a small distance from the jet axis and the most turbulent region about 4 mm from the plasma torch nozzle. Signs of short-time collective rotational movement of the jet core were also identified.