Complex Hermitian matrices have diagonal matrices that are by definition purely real. In addition, some complex triangular matrices computed by LAPACK routines are defined by the algorithm to have real diagonal elements - in Cholesky or QR factorization, for example.
If such matrices are supplied as input to LAPACK routines, the imaginary parts of the diagonal elements are not referenced, but are assumed to be zero. If such matrices are returned as output by LAPACK routines, the computed imaginary parts are explicitly set to zero.