What will happen if I multiply a ket vector by a complex number?

The states of a quantum system, the kets, are elements of a complex Hilbert space (modulo a phase). A complex Hilbert space is nothing more than a fancy (in)finite dimensional vector space equipped with an inner product. So being the Hilbert space a vector space, all the rules which define a vector space apply. The field upon which the vector space is constructed is the one of complex numbers and so to every vector $$|\psi\rangle\in\mathcal{H}$$ there is another vector $$c|\psi\rangle\in\mathcal{H}$$ where $$c\in\mathbb{C}$$ since in vector spaces we can multiply any element by an element of the underlying field, which in this case is the complex numbers.

Think about this like that,- when you multiply a vector by a scalar, you just re-scale original vector. Same idea applies here to vectors in Hilbert space, aka. ket. You just re-scale Hilbert vector by a complex number, an hence get another ket in same Hilbert space.