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