One significant application of almost maximum distance separable (briefly, AMDS) codes is in secure communication systems, such as secure messaging and encrypted data transmission. By incorporating AMDS codes into the data encoding process, information can be safeguarded from accidental errors that might occur during transmission or storage. In this paper, we study all AMDS constacyclic codes of lengtsh p ovepmr F . We also provide some examples of AMDS constacyclic codes over finite fields. As an application, we establish all quantum AMDS (briefly, qAMDS) codes from repeated-root codes of prime power lengths over finite fields using the CSS and Hermitian constructions.