This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next.
It looks to me, that complex-analytic geometry has lost its relative positions since 50's, especially if we compare it to scheme theory. Are there internal mathematical reasons for why that happened?
As we all know, lot's of techniques which were later adapted by algebraic geometrers were originally developed in the complex analytic setting (sheaves, local algebra machinery, etc.). In the 50's, Serre, Cartan, Grothendieck and others seemed to have been developing scheme theory somewhat in parallel with complex-analytic spaces (results on coherent sheaves, base change and cohomology theorems, etc.). But already in the 60's it seems like complex analytic geometry started to lag behind - Grothendieck developed Hilbert scheme in 61, and it took already 5 years for Douady to build a complex-analytic analogue. Then in the 80's came Fulton's monograph on intersection theory, and as far as I can tell intersection theory for complex analytic spaces is still far from the polished form of Fulton. Finally, there is a lot of foundational algebraic work on stacks, and it seems that the amount of publications on complex-analytic stacks is far smaller.
And in general, my impression is that the amount of people doing foundational work in complex analytic geometry is minuscule compared to algebraic geometry (in the recent years I've only come across works of Mauro Porta about the derived complex-analytic set-up and analytic stacks), compared to how active it was before 1990's (there was a nice summary of results by Grauert and Remmert here going back to 1996, showing how active the field was).
Edit: 1. As others have pointed out in the comments, I should have been more specific about what I meant by complex-analytic geometry, since areas like Kahler geometry are very active right now. I basically meant that fundamental side of complex analytic geometry, which shared machinery with algebraic geometry in the 50-60's.
- The point of asking this question at all is (besides curiosity) to learn whether there are some well-understood limitations to the subject.