  We study the non-Hermitian extension of the Lagrangian of the Standard Model
extended by singlet right-handed heavy neutrinos. The Yukawa coupling matrices
comprise of hermitian and non-hermitian components and the neutrino mass
eigenvalues are calculated for three generation case. The increased number of
unknown parameters in the theory due to non-hermitian nature of coupling
matrices impose problem for its productiveness. Hence we apply four zero
texture for the Yukawa matrices. We consider the normal hierarchy of neutrino
mass ordering and find the mixing angles and Dirac CP violating phase by
diagonalizing the lepton mass matrices. The parameters of the mixing matrix are
obtained in terms of the parameters of the non-hermitian Yukawa coupling
matrix. The values of the angles and phase are compatible with experimental
data available for neutrino mass observables and using this the non-hermitian
parameters are constrained. Also, we study the constraints imposed by $CP$
violation in the lepton sector and leptogenesis on the model parameters.
