The instability of the price dynamics of the energy market from a theoretical point of view indicates the inadequacy of the dominant paradigm of the quantitative description of pricing processes, and from a practical point of view, it leads to abnormal shocks and crashes. A striking example is the COVID-stimulated spring drop of spot prices for crude oil by 305% to $36.73 a barrel. The theory of complex systems with the latest complex networking achievements using pragmatically verified econophysical approaches and models can become the basis of modern environmental science. In this case, it is possible to introduce certain measures of complexity, the change in the dynamics of which makes it possible to identify and prevent characteristic types of critical phenomena. In this paper, the possibility of using some econophysical approaches for quantitative assessment of complexity measures: (1) informational (Lempel-Ziv measure, various types of entropies (Shannon, Approximate, Permutation, Recurrence), (2) fractal and multifractal (Multifractal Detrended Fluctuation Analysis), (3) recurrent (Recurrence Plot and Recurrence Quantification Analysis), (4) Lévy’s stable distribution properties, (5) network (Visual Graph and Recurrence based) and (6) quantum (Heisenberg uncertainty principle) is investigated. Each of them detects patterns that are general for crisis states. We conclude that these measures make it possible to establish that the socially responsive exhibits characteristic patterns of complexity and the proposed measures of complexity allow us to build indicators-precursors of critical and crisis phenomena. Proposed quantitative measures of complexity classified and adapted for the crude oil market. Their behavior in the face of known market shocks and crashes has been analyzed. It has been shown that most of these measures behave characteristically in the periods preceding the critical event. Therefore, it is possible to build indicators-precursors of crisis phenomena in the crude oil market.