I recently came across a problem for the extension of the covariant derivative in a associated complex vector bundle. Let $(P, M, \pi, G)$ be a smooth bundle with connection $\omega$ and $W$ be a complex vector space of finite dimension with representation $\rho: G \rightarrow GL(W)$. Then, the associated bundle is $(E = P \times_{\rho}W, M, \pi_{\rho})$. The book by Loring W. Tu defines the covariant derivative only for the case where W is real, as, for $\phi \in \Omega^{k}(P, W)$,
$$D\phi_{p}(X_{0},..., X_{k}) := d\phi_{p}(X_{0}^h,..., X_{k}^h).$$ And the formula, if $\phi \in \Omega^{k}_{\rho}(P, W)$ (right invariant with respect to $\rho$ and horizontal)
$$D\phi = d\phi + \omega \wedge_{\rho} \phi,$$ also for the case where $W$ is real. If we call the elements $E$ the covariant of $[p, v]$, then, for a section $\sigma \in \Omega^{0}(U \subset M, E)$, $\sigma(q) = [s(q), w(q)]$ for $s \in \Omega^{0}(U, P)$ and $w \in C^{\infty}(U, W)$, the covariant derivative induced is
$$D \sigma = [s, dw + \rho_{*}w].$$
However, I have seen many uses of these results where $W = \mathbb{C}$.
Are these results valid for the case where W is complex? If so, does anyone know any text that proves this equation for this case?
Thanks in advance.