Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value $\mu$ and satisfying $|X_i| \le b$. Let $\epsilon > 0$. Then $$\Pr\left\{\left| \frac{1}{K} \sum_{i=1}^K X_i - \mu \right| > \epsilon \right\} \le 2 e^{-K \epsilon^2 / 2 b^2}.$$

I am interested in a generalization where the $X_i$ are instead complex valued. What modifications need to be made to the right hand side of the above inequality? I can get $4e^{-K \epsilon^2 / 4 b^2}$ by applying the real-valued version to each of the real and imaginary parts, but can I do better? I found an article on vector valued martingales that can be applied to give $2e^2 e^{-K \epsilon^2 / 2 b^2}$ if $\mu=0$ but I don't want to make that assumption (I can shift the $X_i$ by the mean to make $\mu=0$, but this would incur the penalty of increasing $b$).