Assuming, you want to 'exclude all duplicate values on both indices', this should work
unique_indices = np.setdiff1d(np.unioin1d(old_close.index.to_list(), new_close.index.to_list()),
np.intersect1d(old_close.index.to_list(), new_close.index.to_list()))
merged_close = pd.concat([old_close, new_close]).ix[unique_indices]
EDIT: Updated unique indices calculation. All duplicate indices are dropped now