A complex intuitionistic fuzzy set is a generalization framework to characterize several applications in decision making, pattern recognition, engineering, and other fields. This set is considered more fitting and coverable to Intuitionistic Fuzzy Sets (IDS) and complex fuzzy sets. In this paper, the abstraction of ( ${\alpha }_{1,2},{\beta }_{1,2}$) complex intuitionistic fuzzy sets and ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroups were introduced regarding to the concept of complex intuitionistic fuzzy sets. Besides, we show that ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. Also, each of ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy normal subgroups and cosets are defined and studied their relationship in the sense of the commutator of groups and the conjugate classes of group, respectively. Furthermore, some theorems connected the ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroup of the classical quotient group and the set of all ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy cosets were studied and proved. Additionally, we expand the index and Lagrange's theorem to be suitable under ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroups.

Modern technology makes it easier to store datasets, but extracting and isolating useful information with its full meaning from this data is crucial and hard. Recently, several algorithms for clustering data have used complex fuzzy sets (CFS) to improve clustering performance. Thus, adding a second dimension (phase term) to the range of membership avoids the problem of losing the full meaning of complicated information during the decision-making process. In this research, the notion of the complex shadowed set (CSHS) was introduced and considered as an example of the three region approximations method simplifying processing with the support of CFS and improving the representation of results attained within. This notion can be founded by extending the shadowed set codomain from $\left\{0,\left[0,1\right],1\right\}$ into $\left\{0e^{i\theta },\left[0,1\right]e^{i\theta },1e^{i\theta }\right\}$. The significance of CSHS was illustrated by giving an example. Additionally, some properties of the CSHS were examined. The basic CSHS operations, complement, union, and intersection were investigated with their properties. Finally, an application in decision-making was illuminated to support the present notion.