Since you asked for "comments, suggestions" here are two thoughts off the top of my head:

1. In Torsion Homologique et Sections Rationnelles by Grothendieck (Séminaire Chevaliey, 1958), the case for principal bundles is worked out. In particular, for $G$-principal bundles over $\mathbb{C}$ where $G$ is isomorphic to a product of groups of type $SL_n$ or $Sp_n$, if the (algebraic) bundle is locally trivial in the analytic topology then it will be locally trivial in the Zariski topology.
2. To learn how to "work around" the difference for cohomological computations, see the propositions in Section 2 in Hodge-Deligne polynomials of $SL(2,\mathbb{C})$-character varieties for curves of small genus by Logares, Muñoz, and Newstead.

Since you asked for "comments, suggestions" here are two thoughts off the top of my head:

1. In Torsion Homologique et Sections Rationnelles by Grothendieck (Séminaire Chevalley, 1958), the case for principal bundles is worked out. In particular, for $G$-principal bundles over $\mathbb{C}$ where $G$ is isomorphic to a product of groups of type $\operatorname{SL}_n$ or $\operatorname{Sp}_n$, if the (algebraic) bundle is locally trivial in the analytic topology then it will be locally trivial in the Zariski topology.
2. To learn how to "work around" the difference for cohomological computations, see the propositions in Section 2 in Hodge polynomials of $\operatorname{SL}(2,\mathbb{C})$-character varieties for curves of small genus by Logares, Muñoz, and Newstead.