Where is the world of Imaginary numbers? Complex numbers have two parts, Real and Imaginary parts.
Real world is base of Real numbers.
but 
where is (or what is) the world of Imaginary numbers?

I would say that complex numbers are as real as physical measurements in the real world, because of at least one peculiar thing called quantum mechanics, in which the wave-function for any particle is a differentiable complex-valued function. The square of the absolute value of that function is essentially the density function, and the argument is quite like a phase, which can be measured somewhat by getting two particle with different phase to interact and produce an interference pattern.
Also, to partially answer the related question of whether there is a reason for using complex numbers instead of vectors or some other equivalent objects, there is at least one compelling reasons. Firstly $\mathbb{C}$ is the essentially unique algebraic closure of $\mathbb{R}$, which is the completion of $\mathbb{Q}$, which is the natural extension of $\mathbb{Z}$, which is indisputably real because we can conceive of doing something once, twice, ..., or undoing it once (-1), twice (-2), ... or doing nothing at all (0). The algebraic closure of $\mathbb{R}$ must contain a square-root of $-1$, and it is so beautiful that when you add just that one element it becomes the algebraic closure!
