This is basically a request for a graduate course, though. Definitely not a first course in linear algebra.
TBH, I don’t think there are surprises in the “general case” (whatever that is… modules over a PID??) that you can’t see by understanding the real&complex situation.
No, but a second course. Still firmly in undergraduate territory, second semester at my university. Most undergraduate linear algebra textbooks also contain this, unless they are matrix computation heavy (usually for the first course).
TBH, I don’t think there are surprises in the “general case” (whatever that is… modules over a PID??) that you can’t see by understanding the real&complex situation.