Dirac argues on page 17 of his book, The Principles of Quantum Mechanics, that multiplication of a ket by a complex number shouldn't change the state this ket represents. But then concludes:
Thus a state is represented by the direction of a ket vector, and any length one may assign to the ket vector is irrelevant.
Bearing in mind that he has only stated that we need to make a generalization of vectors in a space of an infinite number of dimensions, how is he able to conclude this?
Obviously I cannot know what Dirac is thinking, but I think it is just that his direction does not correspond exactly to your direction.
We "know", just as Dirac does, that quantum states are members of some Hilbert space $\mathcal{H}$. We also know that scalar multiplication should not change the state, so $\lvert \psi \rangle$ and $\lambda \lvert \psi \rangle$ should represent the same state for $\lambda \in \mathbb{C} / \{0\}$.
The ray in the Hilbert space (or the direction) given by a single vector $\lvert \psi \rangle$ of a state is $R_\psi = \{\lambda \lvert \psi \rangle \vert \lambda \in \mathbb{C} / \{0\}\}$. Dirac wants us, since all $\lvert \phi \rangle \in R_\psi$ represent the same state, to declare the equivalence relation $\lvert \psi \rangle \sim \lvert \phi \rangle \text{ iff } \lvert\phi\rangle \in R_\psi$ on $\mathcal{H}$ and obtain the projective Hilbert space as $\mathbb{P}\mathcal{H} := \mathcal{H} / \sim$.
It should be noted that, in "intuitive" terms, the complex ray $R_\psi$ is like a punctured plane created by taking the naive real ray $r_\psi := \{c\lvert \psi \rangle\ \vert c \in \mathbb{R^+}\}$ and rotating it around the origin as $R_\psi = \bigcup_{k \in [0,2\pi)} (\mathrm{e}^{\mathrm{i}k} r_\psi)$
Stated in a simpler way, kets are the generalisation of vectors to complex and potentially continuous/infinite dimension space (Hilbert spaces). Yet you can keep in mind the image of a vector to begin with.
When you multiply a vector by a nonzero real number, its direction does not change. The same is valid for vectors from a Hilbert space. If you multiply them by a nonzero complex number, their "direction" remains the same.
Regarding the change in physical information, it is unchanged too for the following reason. In state $|\psi>$, the expectation value of quantity $A$ is $\frac{<\psi|A|\psi>}{<\psi|\psi>}$. If you multiply $|\psi>$ by any nonzero complex number, you can check that the expectation value does not change. It's the same physics.
A physical state is defined by the density matrix, so, if you define the density matrix by :
$\rho = \frac{|\psi\rangle \langle \psi|}{\langle \psi| \psi\rangle}$
it is easy to see that any multiplication by a complex number does not change the density matrix, so does not change the physical state.
It is probably what Dirac means, while I have not the book.
