  This paper explores quantum field theories with pseudo-Hermitian
Hamiltonians, where PT-symmetric Hamiltonians serve as a special case. In
specific regimes, these pseudo-Hermitian Hamiltonians have real eigenspectra,
orthogonal eigenstates, and unitary time evolution. So far, most
pseudo-Hermitian quantum field theories have been constructed using analytic
continuation or by adding non-Hermitian terms to otherwise Hermitian
Hamiltonians. However, in this paper, we take a different approach. We
construct pseudo-Hermitian scalar and fermionic quantum field theories from
first principles by extending the Poincar\'e algebra to include non-Hermitian
generators. This allows us to develop consistent pseudo-Hermitian quantum field
theories, with Lagrangian densities that transform appropriately under the
proper Poincar\'e group. By doing so, we establish a more solid theoretical
foundation for the emerging field of non-Hermitian quantum field theory.
