If the space has a mixed metric signature, not all the basis vectors are Hermitian. Nevertheless, they are defined to be self-adjoint under reversion. The vector transpose conjugate is, therefore, not necessarily the same as the Hermitian conjugate of its matrix representation. This distinction becomes important when considering Lorentz transformations in Minkowski Space. (Classical Mechanics - J. Michael Finn)
What does "basis vectors being Hermitian" mean?
And how can vector transpose conjugate differ from hermitian conjugate?