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$$ (\psi_{a}^{\ast} \gamma^0_{ab} \chi_{b})^{\ast} = \psi_{a} \gamma^{0\ast}_{ab} \chi_{b}^{\ast} $$ is not correct. Rather $$ (\psi_{a}^{\ast} \gamma^0_{ab} \chi_{b})^{\ast} =\chi_{b}^{\ast} \gamma^{0\ast}_{ab} \psi_{a}= - \psi_{a} \gamma^{0\ast}_{ab} \chi_{b}^{\ast} $$ Note that the Hermitian is defined as $$ (AB)^\dagger = B^\dagger A^\dagger $$ Note ...


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