The concept of complex fuzzy set (CFS) and complex intuitionistic fuzzy set (CIFS) is two recent developments in the field of fuzzy set theory. The significance of these concepts lies in the fact that these concepts assigned membership grades from the unit circle in the plane, i.e., in the form of a complex number instead of [0,1] interval. CFS cannot deal with information of yes and no type, while CIFS works only for a limited range of values. To deal with these kinds of problems, in this article, the further development of the theory of complex Pythagorean fuzzy sets (CPFSs) has been discussed. Some new operations on CPFSs, such as bounded difference, disjoint sum, disjunctive sum, probabilistic sum, bold sum, bold intersection, s-norm and t-norm have been proposed. The distance of two CPFSs has been proposed. This distance measure is then used to define \(\eta \)-equalities of CPFSs. Moreover, we define some new notions for the multicriteria decision-making (MCDM) problems such as the CPF decision-making matrix, CPF \(\max \) and \(\min \) decision-making matrix, and the distance measure between CPF \(\max \) and \(\min \) decision-making matrices. An MCDM method has been developed on the proposed novel notions. A real-life example demonstrates that the MCDM method developed in the paper can be utilized to deal with problems of uncertainty. Further, the comparative study of CPFSs with complex intuitionistic fuzzy sets, Pythagorean fuzzy sets, intuitionistic fuzzy sets, complex fuzzy sets, and fuzzy sets is established.

复杂模糊集（CFS）和复杂直觉模糊集（CIFS）的概念是模糊集理论领域的两个最新发展。这些概念的意义在于，这些概念从平面中的单位圆来分配隶属度，即以复数的形式而不是[0,1]区间。CFS 无法处理是和否类型的信息，而 CIFS 仅适用于有限范围的值。为了解决这类问题，本文讨论了复杂勾股模糊集 (CPFS) 理论的进一步发展。已经提出了一些关于 CPFS 的新操作，例如有界差分、不相交和、析取和、概率和、粗和、粗交、s-范数和 t-范数。已经提出了两个 CPFS 的距离。\(\eta \) -CPFS 的等式。此外，我们为多准则决策（MCDM）问题定义了一些新概念，例如 CPF 决策矩阵、CPF \(\max \)和\(\min \)决策矩阵，以及之间的距离度量CPF \(\max \)和\(\min \)决策矩阵。已根据提出的新颖概念开发了 MCDM 方法。一个真实的例子表明，本文开发的 MCDM 方法可用于处理不确定性问题。进一步，建立了CPFS与复杂直觉模糊集、毕达哥拉斯模糊集、直觉模糊集、复杂模糊集、模糊集的比较研究。