The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention over the last years. Some efforts have been also devoted to analyze fractional maps with special features. This paper makes a contribution to the topic by introducing a new fractional map that is characterized by both particular dynamic behaviors and specific properties related to the system equilibria. In particular, the conceived one dimensional map is algebraically simpler than all the proposed fractional maps in the literature. Using (...) numerical simulation, we investigate the dynamic and complexity of the fractional map. The results indicate that the new one-dimensional fractional map displays various types of coexisting attractors. The approximate entropy is used to observe the changes in the sequence sequence complexity when the fractional order and system parameter. Finally, the fractional map is applied to the problem of encrypting electrophysiological signals. For the encryption process, random numbers were generated using the values of the fractional map. Some statistical tests are given to show the performance of the encryption. (shrink)