$\|C(k)\|$ can be arbitrarily large, since e.g. you can add some large integer multiple of $2\pi i I$ without changing $e^{C(k)}$. If you want to avoid this, you might specify that $C(k)$ is the principal branch of the logarithm of $e^{kA+B} e^{-kA}$ (note that although $e^{kA+B} e^{-kA}$ is not hermitian, it has the same eigenvalues as $e^{-kA/2} e^{kA+B} e^{-kA/2}$ which is positive definite)

$\|C(k)\|$ can be arbitrarily large, since e.g. you can add some large integer multiple of $2\pi i I$ without changing $e^{C(k)}$. If you want to try to avoid this, you might specify that $C(k)$ is the principal branch of the logarithm of $e^{kA+B} e^{-kA}$ (note that although $e^{kA+B} e^{-kA}$ is not hermitian, it has the same eigenvalues as $e^{-kA/2} e^{kA+B} e^{-kA/2}$ which is positive definite). Then the eigenvalues of $C(k)$ will be $O(k)$; unfortunately, since $C(k)$ is not hermitian or normal this does not imply $\|C(k)\| = O(k)$.