A complex intuitionistic fuzzy set is an innovative uncertainty set whose membership and non-membership functions take values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a new distance measure for complex intuitionistic fuzzy sets. The distance of two complex intuitionistic fuzzy sets measures the difference between the grades of two complex intuitionistic fuzzy sets as well as that between the phases of the two complex intuitionistic fuzzy sets. This distance measure is then used to define (α, β)-equalities of complex intuitionistic fuzzy sets which coincide with those of intuitionistic fuzzy sets already defined in the literature if complex intuitionistic fuzzy sets reduce to traditional intuitionistic fuzzy sets. Two complex intuitionistic fuzzy sets are said to be (α, β)-equal if the distance between their membership degree is less than 1 −α and the distance between their non-membership degree is less than β. Meanwhile we shows how various operations between complex intuitionistic fuzzy sets affect given (α, β)-equalities of complex intuitionistic fuzzy sets. Finally, complex intuitionistic fuzzy relations are discussed and some examples are given to illuminate the results obtained in this paper.