Physical concept of imaginary energy I want to know that what's the physical description of complex energy? The energies have the real and imaginary parts. 
 A: The imaginary part of an energy tells you the lifetime of a state.  The amplitude to detect a particle at time t with a complex mass $m=m_R+im_I$ evolves in time as
$$
e^{i\frac{mt}{\hbar}}=e^{i\frac{(m_R+im_I)t}{\hbar}}=e^{i\frac{m_R t}{\hbar}}e^{\frac{-m_I t}{\hbar}}
$$
Therefore, the lifetime of the particle is $\tau=\frac{\hbar}{m_I}$. Another example, in colliding $e^+e^- \rightarrow \psi \rightarrow e^+e^-$ the scattering amplitude as a function of the center of mass energy $E_{cm}$ of the $e^+e^-$ has a pole $$ScatteringAmplitude \approx \frac {1}{E_{cm}-m_{\psi}}$$  In the complex energy plane $E_{cm}$ moves along the real axis and the complex mass $m_{\psi}$ is above the axis because of its imaginary part.  The farther $m_{\psi}$ is above the real axis, the broader the resonance is as $E_{cm}$ moves by, and the shorter the lifetime of the particle $\psi$.
As @John Rennie commented, the energy eigenvalues of a Hermitian Hamiltonian H are real. The H is Hermitian because $e^{i\frac{Ht}{\hbar}}$ is unitary (ie: the sum of the probabilities to be in some eigenstate always remains 1). An eigenstate of a Hermitian H does not go away with time. The particle $\psi$ with complex mass is not an eigenstate of this Hermitian H. 
A: By definition in both classical and quantum mechanics the energy cannot be a complex number.
In classical mechanics, given a Lagrangian $L = T-V$, the energy is defined as 
$$
E_{L} = v\frac{\partial L}{\partial v} - L
$$
one could make an argument for the Lagrangian to not necessarily be the difference between kinetic and potential term; nevertheless it will always be a quadratic form of real functions of position and velocities because the equations of motion must be of second order.
In quantum mechanics the energy is the expectation value of a Hermitian operator, therefore it is a real number. The original operator you take the expectation value of must by all means by Hermitian due to the conservation of the norm for states in quantum mechanics (or conservation of probability, or however else you want to call it).
