Complex analytic vs algebraic geometry

Though I am not an expert on this I think that the shift toward algebraic geometry is not entirely sociological. Consider the following statement which is true in both the category of schemes and analytic spaces :

The push-foward of a coherent sheaf by a proper map is coherent.

In algebraic geometry, this statement is rather a routine-like statement, once you have the tools crafted by Grothendieck. In the complex-analytic setting, this is a hard theorem (due to Grauert and Remmert) and no simple proof of it is known.

Another result one could be interested is Mori's bend-and-break lemma. It is probably one of the most important tool in modern birationnal geometry (and was celebrated as one of the most important results in algebraic geometry in the late 70's early 80's). The original proof goes through caracteristic $p$ techniques (namely the use of the Froebenius morphism to amplify a given vector bundle).

Siu and others have claimed to have a purely analytical proof of the bend-and-break lemma. But as far as I can tell (I am not an expert, but I know a bit of birational geometry) their analytical proofs are very hard to follow.

In my opinion, many people in the 70's and 80's left analytical geometry to embrace algebraic geometry because so many powerful and beautiful results had simple and crystalline (in the non-Grothendickian sense) proofs in the algebraic category while their analytic counterparts looked, at the time, either out of reach or extremely hard to prove.

A very down-to-earth baby example at the undergraduate level is the following:

A regular function on the affine line which has a non-isolated zero vanishes everywhere.

In the algebraic category, there is a one-line proof using Euclidean division. In the analytic setting (that is for analytic functions over $\mathbb{C}$), you have to work a little to prove this.

Note however that the Empire may be striking back. Indeed, outside of Siu's and Demailly's circles, which are, in my opinion, not very active anymore, there is a new approach to the minimal model program using the Ricci-flow and the techniques Perelman introduced to prove the Poincaré conjecture. In a word, the new idea is to start with a (smooth) variety whose canonical bundle is not nef, run the Ricci flow on it and hope that it will converge to a minimal model. So, at least as far as birational geometry is concerned, it seems that analysis is back!

I'm not sure this question is a good fit form mathoverlow, but here are a few thoughts. I'll probably delete this answer in a while.

1. Let me elaborate my comment concerning sociology a bit further. A (sub)field (not necessarily in mathematics) becomes "hot" when exciting new techniques are introduced that resolve old problems, and create new directions. But at some point it seems to "cool", at least when observed from a distance, when some of these have played out. This seems to have happened with several complex variables in the period of the late 1940's- 1960's with the introduction, and subsequent development, of coherence and other sheaf theoretic ideas. After that I have the sense, from my colleagues who work in the area, that the sorts of the questions that people in SCV think about are further away from the interests of algebraic geometers, and so are perhaps less visible to us.
2. If by "complex analytic geometry", one includes areas such as complex differential geometry etc. then the subject is very much alive and very active, with many interactions with algebraic geometry.

For what regards Intersection Theory in Analytic Geometry, in my (nonexpert!) view it is important to take a look at the work of D. Barlet on algebraic cycles on analytic spaces. In his paper “Espace analytique r´eduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie”, pp. 1–158 in Fonctions de plusieurs variables complex, Springer Lecture Notes in Mathematics, He shows as the set of compact $n$-cycles $\mathcal C_n(X)$ of a reduced complex space $X$ carries the structure of complex reduced space. Moreover, He also proves that this space coincides with its algebro-geometric counterpart, the Chow monoid. In the loc. cit. paper, one of the most interesting thing is the introduction of an Intersection Theory for cycles on a complex reduced space. It turns out (not really easy to prove) that the subset $\Omega\subseteq \mathcal C_n(X)^{\times 2}$ consisting of cycles intersecting properly is "really big" and that the intersection map (where defined) has nice properties. Recently, D. Barlet and J. Magnusson also published (Springer) the first of a two-volume series Complex Analytic Cycles.