I mentioned this in a sister comment, but I will add:

* Schenkerian analysis is probably the modern method of looking at Bach. A sister comment also indicates that Schenker was a controversial figure (he was a German fascist), but I believe the follow-ups from other theorists on his methods are the modern ones to use for analysis.

* Analyzing otherwise in the context of counterpoint and intervals - see Gradus ad Parnassum and Kennan's book on counterpoint.

I guess you could use figured bass notation. It's more complex than modern chord notation because it tells you which inversion to use, and it notates secondary dominants (e.g. V of V, which you'd write simply as D7 in jazz notation if you're in C). Personally I know too little about it to use it effectively and therefore I would stick with modern chord notation even when talking about classical pieces.