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 shceme 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 work in progress.
And in general, my impression is that the amount of people doing complex analytic geometry is minuscule compared to algebraic geometry (I only know of big research groups around Demailly in Europe and Siu in the US).
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 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.