Another example, actually a lot more important, is to take a Hartogs figure, such at 
\[
X:=D\times \{0\}\cup \partial D \times D \subset C^2
\] 
which your are welcome to think of as $R^4$. 

Take a neighborhood $U$ of $X$.  Then the first cohomoloy of a small neighborhood $U$ with values in the sheaf of holomorphic functions is not zero.

I could spell t out if you are interested.

John Hubbard