Just off the coast to the southwest is the Île de Mogador, which has some interesting structures. It's actually two islands and several tiny islets – also known as the famed Îles Purpuraires (Purple Isles) of antiquity. The uninhabited islands are a sanctuary for Eleonora's falcons, which can also be easily seen through binoculars from Essaouira beach.
It is possible to arrange a private boat trip to the islands outside the breeding season, but you need to obtain a permit (free) from the port office; with that in hand, head for the small fishing boats to negotiate the trip out there. If you want to stay a few hours, fix a time for the boat to come and pick you up.
