This work studies Schrodinger equation involving a complex potential. The Hamiltonian is non- Hermitian, non- PT symmetry and non-pseudo-Hermitian. We solve three such kinds of Hamiltonian, including an imaginary potential hump in an infinite well, the imaginary barrier potential by itself, and the imaginary delta potential. All the properties and techniques in quantum mechanics can be bought out to examine if they can be generalized to such a complex potential, for instance, can the standard perturbation theory be applied to a complex perturbation, is there an analogy to the Fano resonance, do the eigenfunctions form a complete basis, or if the path-integral formalism still valid, etc.

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