So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd like to fix this.
I've recently read a section in Bredon's "Geometry and Topology", where he defines differential forms on real manifolds. However, most of the differential forms I work with are complex differential forms, and I understand there are some additional subtleties involved coming from the complex structure. I'd like to understand exactly how complex differential forms compare with real differential forms, and especially how to see them as special cases of real differential forms with additional structure.
Note that I've never taken a course in differential geometry, so if possible I'd like a resource that's as explicit as possible while still using the most general and precise language.