TITLE: Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications
QUESTION [60 upvotes]: If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications.  I've heard informal examples (often involving moduli spaces), but am not sure of the references.  

What are the simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytificaitons?

By "simplest", I mean by one of the following measures.

(best)  An example whose proof is as elementary as possible, and ideally short.  This of course requires proof that the complex algebraic varieties are nonisomorphic, and that the analytifications are isomorphic.
A known example that is simple to state, but may have a complicated proof.  (Ideally there should be a reference.)
An expected, folklore, or conjectured example.

REPLY [3 votes]: If you look at local rings, it is easy to construct such examples. For example, if $X,Y$ are smooth of same dimension, then for any point $x\in X,y\in Y$, the completions of $O_{X,x}, O_{Y,y}$ are isomorphic, but of course the algebraic local rings are not necessarily isomorphic (for example, if $X,Y$ are not birational, then even their fraction fields are not isomorphic.)