In the dirac equation we have a term which is proportional to [tex] \alpha p [/tex]. In the book they say that [tex] \alpha [/tex] must be an hermitian operator in order for the Hamiltonian to be hermitian. As I understand, we require this because we want [tex] (\alpha p)^\dagger = \alpha p[/tex]. But [tex] (\alpha p)^\dagger = p^\dagger \alpha^\dagger = p \alpha [/tex], and so the order of the operators still change. So if we just require both operators to be hermitian their product will still change if we take the hermitian conjugate.