This paper presents a complex generalization of quantum mechanics where the standard Hermitian Hamiltonian is replaced by a complex Hamiltonian that satisfies space-time reflection symmetry (PT symmetry). The authors show that such Hamiltonians can have real and positive spectra, and that they can be used to construct a consistent quantum theory with unitary time evolution. The key idea is that PT symmetry ensures the reality of the spectrum, and that a new symmetry, C, can be introduced to define a positive definite inner product. This inner product is constructed using CPT conjugation, and allows for the definition of observables with real eigenvalues and unitary dynamics. The paper also discusses the implications of this approach for quantum field theory, showing that it can lead to new types of quantum field theories with self-interaction potentials such as -igφ³ or -gφ⁴. The authors conclude that their approach is a natural extension of conventional quantum mechanics, replacing the mathematical condition of Hermiticity with the physical condition of PT symmetry and CPT invariance. The paper also discusses the implications of this approach for the structure of Hilbert spaces and the nature of quantum states in complex quantum theories.This paper presents a complex generalization of quantum mechanics where the standard Hermitian Hamiltonian is replaced by a complex Hamiltonian that satisfies space-time reflection symmetry (PT symmetry). The authors show that such Hamiltonians can have real and positive spectra, and that they can be used to construct a consistent quantum theory with unitary time evolution. The key idea is that PT symmetry ensures the reality of the spectrum, and that a new symmetry, C, can be introduced to define a positive definite inner product. This inner product is constructed using CPT conjugation, and allows for the definition of observables with real eigenvalues and unitary dynamics. The paper also discusses the implications of this approach for quantum field theory, showing that it can lead to new types of quantum field theories with self-interaction potentials such as -igφ³ or -gφ⁴. The authors conclude that their approach is a natural extension of conventional quantum mechanics, replacing the mathematical condition of Hermiticity with the physical condition of PT symmetry and CPT invariance. The paper also discusses the implications of this approach for the structure of Hilbert spaces and the nature of quantum states in complex quantum theories.