  The complex-valued quantum mechanics considers quantum motion on the complex
plane instead of on the real axis, and studies the variations of a particle
complex position, momentum and energy along a complex trajectory. On the basis
of quantum Hamilton-Jacobi formalism in the complex space, we point out that
having complex-valued motion is a universal property of quantum systems,
because every quantum system is actually accompanied with an intrinsic complex
Hamiltonian originating from the equation. It is revealed that the conventional
real-valued quantum mechanics is a special case of the complex-valued quantum
mechanics in that the eigen-structures of real and complex quantum systems,
such as their eigenvalues, eigenfunctions and eigen-trajectories, are invariant
under linear complex mapping. In other words, there is indeed no distinction
between Hermitian systems, PT-symmetric systems, and non PT-symmetric systems
when viewed from a complex domain. Their eigen-structures can be made
coincident through linear transformation of complex coordinates.
