A classical real scalar field admits a decomposition $$\phi(x)\sim a_pe^{-ip\cdot x}+a_p^*e^{+ip\cdot x}$$ which tells that at each $x$, there exists a real number i.e., one degree of freedom at each spacetime point $x$.
A classical complex scalar field can be decomposed as $$\phi(x)\sim a_pe^{-ip\cdot x}+b_p^*e^{+ip\cdot x}$$ which implies that it assigns a complex number i.e. two real degrees of freedom to each spacetime point.
Let us now look at a Dirac field. It can be decomposed as $$\Psi(x)\sim \sum\limits_{s=1,2}\Big[a^s_pu^s_pe^{-ip\cdot x}+b^{*s}_pv^s_pe^{+ip\cdot x}\Big].$$
How many real degrees of freedom do I have now at a given spacetime point?
If we pretend for a moment that $u_p^s$ and $v_p^s$ are complex numbers instead of column vectors, it looks like that we have four real degrees of freedom at each $x$: $\Psi(x)$ is made up of two independent complex numbers for each value of $s$.
Is my counting right? Do I make a mistake by treating $u$ and $v$ to be numbers? But I seem to get a problem if I start putting spinor indices.
