= Solution
For <Hermitian matrices>, $A^\dagger=A$ and $B^\dagger=B$, so
$$
[A,B]^\dagger=B^\dagger A^\dagger-A^\dagger B^\dagger=-[A,B].
$$
The <commutator> is an <anti-Hermitian matrix>. Consequently
$$
\boxed{\alpha=i,\qquad C=-i[A,B]}
$$
makes $C^\dagger=C$. More generally, any nonzero purely imaginary $\alpha$ works. For a nonzero <commutator> this is also necessary if $C$ is to be <Hermitian>: comparing the <Hermitian adjoint> equations gives $(\overline\alpha+\alpha)C=0$.
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