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Is the term $(\partial^{\mu}\phi)^{\dagger}$ same as $\partial^{\mu}\phi^{\dagger}$ for any complex scalar field $\phi$?

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It depends on what inner product you are using to define the $\dagger$ operation. If you are including integrals over $x$ as part of the inner product, then $(\partial^\mu)^\dagger = - \partial^\mu$,just as it is with the momentum operator in QM. If $\phi$ is an operator on a Hilbert space and you are only taking the adjoint on that space, then the $\partial^\mu$ is unaffected by $\dagger$. I expect that the latter is what you means as it looks like something you would find in the action for a scalar field.

As always, $\dagger$ needs a specification of the inner product to define it.

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