High-dimensional data structure in Python
If you need labelled arrays and pandas-like smart indexing, you can use xarray package which is essentially an n-dimensional extension of pandas Panel (panels are being deprecated in pandas in future in favour of xarray).
Otherwise, it may sometimes be reasonable to use plain numpy arrays which can be of any dimensionality; you can also have arbitrarily nested numpy record arrays of any dimension.
What is best way to store and analyze high-dimensional date in python? I like Pandas DataFrame and Panel where I can easily manipulate the axis. Now I have a hyper-cube (dim >=4) of data. I have been thinking of stuffs like dict of Panels, tuple as panel entries. I wonder if there is a high-dim panel thing in Python. 
update 20/05/16: 
Thanks very much for all the answers. I have tried MultiIndex and xArray, however I am not able to comment on any of them. In my problem I will try to use ndarray instead as I found the label is not essential and I can save it separately.
update 16/09/16:
I came up to use MultiIndex in the end. The ways to manipulate it are pretty tricky at first, but I kind of get used to it now.