Suppose there is a vector $V = e^{it}$ , and taking $dV/dt$ would just give me $iV$.
What can be concluded from this...
So what I am asking is that what would the presence of $i$ next to $V$ after differentiation mean?
what other operators return the function with an i other than this one? is there a name for this kind of functions? can they be physically or geometrically explained?
It isn't unusual that we write things as complex exponentials. For example the wavefunction of a free particle can be written as:
$$ \psi(x, t) = e^{i(kx - \omega t)} $$
And differentiating this wrt time does indeed give you:
$$ \frac{d}{dt}\psi(x, t) = -i\omega \psi(x, t) $$
however any quantity written in this way cannot be a physical observable. In this case that's fine because we find observables from the wavefunction by evaluating $\langle\psi\vert \hat{O} \vert\psi\rangle$ for the appropriate hermitian operator $\hat{O}$, and the results are always a real number.
So there may well be occasions when it's appropriate to write some property of a physical system as a complex exponential. However that property will never be a physical observable so you don't need to worry about factors of $i$ appearing.
You might be interested to note that your expression for $V$ can be rewritten using Euler's formula as:
$$ V = e^{it} = \cos t + i\sin t $$
So $iV$ is just:
$$ \frac{dV}{dt} = iV = -\sin t + i\cos t $$
So if you take the vector from the origin to the point $V$ in the complex plane you have just rotated this vector by $\pi/2$.
