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The concept of complex Pythagorean fuzzy set <inline-formula> <tex-math notation="LaTeX">$(\mathbf {CPFS})$ </tex-math></inline-formula> is recent development in the field of fuzzy set <inline-formula> <tex-math notation="LaTeX">$(\mathbf {FS})$ </tex-math></inline-formula> theory. The significance of this concept lies in the fact that this theory assigned membership grades <inline-formula> <tex-math notation="LaTeX">$\psi $ </tex-math></inline-formula> and non-membership grades <inline-formula> <tex-math notation="LaTeX">$\hat {\psi }$ </tex-math></inline-formula> from unit circle in plane, i.e., in the form of a complex number with the condition <inline-formula> <tex-math notation="LaTeX">$(\psi)^{2}+ (\hat {\psi })^{2}\le 1$ </tex-math></inline-formula> instead from &#x005B;0, 1&#x005D; interval. This is an expressive technique for dealing with uncertain circumstances. The aim of this study is to proceed the classification of the unique framework of <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFS}$ </tex-math></inline-formula> in algebraic structure that is field theory and examine its numerous algebraic features. Also, we initiate the important examples and results of certain field. Furthermore, we illustrate that every complex Pythagorean fuzzy subfield (<inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSF}$ </tex-math></inline-formula>) generates two Pythagorean fuzzy subfields <inline-formula> <tex-math notation="LaTeX">$(\mathbf {PFSFs})$ </tex-math></inline-formula>. We also prove many useful algebraic aspects of this notion for a <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSF}$ </tex-math></inline-formula>. Moreover, we demonstrate that intersection of two complex Pythagorean fuzzy subfields <inline-formula> <tex-math notation="LaTeX">$(\mathbf {CPFSFs})$ </tex-math></inline-formula> is also <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSF}$ </tex-math></inline-formula>. Additionally, we discuss the novel idea of level subsets of <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSFs}$ </tex-math></inline-formula> and demonstrate that level subset of <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSF}$ </tex-math></inline-formula> form subfield. Additionally, we improve the application of this theory to show the concept of the direct product of two <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSFs}$ </tex-math></inline-formula> is also a <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSF}$ </tex-math></inline-formula> and produce several novel results on direct product of <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSFs}$ </tex-math></inline-formula>. Finally, we explore the homomorphic images and inverse images of <inline-formula> <tex-math notation="LaTeX">$\mathbf {CPFSFs}$ </tex-math></inline-formula>.