The work is devoted to a comparative analysis complexity of traditional stock market indices and social responsible indices in the example Dow Jones Sustainability Indices and Dow Jones Industrial Average. As measures of complexity, the entropies of various recurrence indicators are chosen - the entropy of the diagonal lines of the recurrence diagram, recurrence probability density entropy and recurrence entropy. It is shown that these measures make it possible to establish that the socially responsive Dow Jones index is more complex. A comprehensive assessment of complexity reveals the nature of the effectiveness of social responsible indices and opens up new opportunities for investor risk management.