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