In this paper, we study the connection between generalized quasi-left alter BCI-algebra and commutative Clifford semigroup by introducing the concept of an adjoint semigroup. We introduce QM-BCI algebra, in which every element is a quasi-minimal element, and prove that each QM-BCI algebra is equivalent to generalized quasi-left alter BCI-algebra. Then, we introduce the notion of generalized quasi-left alter-hyper BCI-algebra and prove that every generalized quasi-left alter-hyper BCI-algebra is a generalized quasi-left alter BCI-algebra. Next, we propose a new notion of quasi-hyper BCI algebra and discuss the relationship among them. Moreover, we study the subalgebras of quasi-hyper BCI algebra and the relationships between Hv-group and quasi-hyper BCI-algebra, hypergroup and quasi-hyper BCI-algebra. Finally, we propose the concept of a generalized quasi-left alter quasi-hyper BCI algebra and QM-quasi hyper BCI-algebra and discuss the relationships between them and related BCI-algebra.