Complexfying a five-parameter exponential-type potential with shape invariance, we obtain a new η-pseudo-Hermitian complex potential with PT symmetry, V(x)=(V1/q)tanhq2(αx)−iV2sechq(αx)·tanhq(αx), q>0 and q≠1. The discrete energy eigenvalues are shown to be real when &z.sfnc;V2&z.sfnc;⩽(1/√ of q)((α2q/4)+V1) while they are complex conjugate pairs if &z.sfnc;V2&z.sfnc;>(1/q)((α2q/4)+V1) while they are complex conjugate pairs if &z.sfnc;V2&z.sfnc;>(1/√ of q)((α2q/4)+V1). [Copyright &y& Elsevier]