How many degrees of freedom for the spin 1/2? I count four but most of the documentation I read, says just one.
1) One degree of freedom for the spin measurement outcome, either up or down.
3) three degrees of freedom for the unit quaternion representing the orientation of the spin in space.
Is this correct, or have I made a mistake?
 A: Using the strict definition for 'number of degrees of freedom' you've given in the comments,

the quantity of real numbers required to describe the system,

then a spin-1/2 particle has two degrees of freedom if it is in a pure state. The general state of such a particle can be written as
$$
|\psi⟩ = \cos\left(\frac\theta2\right)\left|↑ \right>+e^{i\varphi}\sin\left(\frac\theta2\right)\left|↓\right>,
$$
where $\left|↑ \right>$ and $\left|↓\right>$ are eigenstates of the $z$ component of the spin vector. Here the state is parametrized in terms of the two real parameters $\theta$ and $\varphi$, which essentially parametrize the Bloch sphere - the natural manifold representation of the state space of the particle.
More importantly, this number is derived as the manifold dimension of $\mathbb C\mathrm P^2$, the complex projective space of dimension $2$, which is essentially the complex vector space $\mathbb C^2$ (the two-complex-dimensional vector space of all the complex linear combinations of $\left|↑ \right>$ and $\left|↓\right>$, so four real dimensions), with one taken out by the fixed normalization and one taken out by the irrelevance of the global phase. More generally, a particle of spin $j$ has a $(2j+1)$-complex-dimensional state space, and therefore can be described using $4j$ real numbers.

That said, the definition you've given is essentially meaningless, and it is essentially incapable of describing systems as simple as a massive spinless particle on a 1D line; there the Hilbert state space is infinite-dimensional (spanned e.g. by the harmonic-oscillator basis) and you absolutely need an infinite number of real numbers to describe any state of the system. Moreover, the same is true in Liouvillian classical mechanics for the same situation.
What that really means is that the term "degree of freedom" is not a particularly useful concept in a quantum mechanical situation, particularly where you might have a finite-dimensional system coupled to an infinite-dimensional one (as you might e.g. in the hugely popular Jaynes-Cummings model and its plethora of generalizations). In the situations where you do need to keep track of such things, you end up needing to specify the types of systems (i.e. the dimension of their state spaces) and their number, and there just isn't a single bag where you can chuck everything in and get a meaningful number.
You find any of that unpalatable? Well, join the back of the queue.
