It is true that the "Al" prefix in Spanish surnames often connotes an Arabic origin: e.g. Alcazar, Almodovar, Alcantara, etc.However, this is not always the rule.Names like Alejandro, Alfonso, Alonso, Alfredo, etc. are not Arabic.
I doubt very much that Alderete is Moorish.Nor is Alderete of French origin as some believe.
Alderete is either Basque in origin (a derivative of Aldaragui) or Asturian (a derivative of Alderedo).