Your question about *hidden things* gets at an interesting point (even though "Under / underneath vs. etc." is [much discussed on ELL](http://ell.stackexchange.com/search?q=underneath)).

First, as you note, under / underneath are equivalent, and underneath can be more formal. Beneath and underneath are slightly more literary or "fancier" words than the simple "under." When thinking about your question on hiding, we can observe that in general:

_-neath words have more **intensity**_. 

That intensity can be used to emphasize a distinction:

> Under the oak tree is a rock. Underneath the oak tree is a treasure chest.

The rock is below the tree branches (on the ground); the chest is below the tree roots (in the ground). This is not always the case -- I can sit beneath a tree and read a book! But when multiple relations come into play, -neath is the more intense term.

It can also convey an emotion:

> He is below me. (Might refer to standing on a ladder, or to rank, or anything)  
> He is beneath me. (Without context this is an insult about social status)

When -neath intensifies "under," that often implies controlling, covering, or hiding.

> They met under the bridge.  
> They met underneath the bridge.

Here "under*neath*" might indicate that the space under the bridge is small, or that the bridge is big and impressive. We are made more aware of the bridge they are meeting under, they are *really* under it.

> Her shoes were under the chair.  
> Her shoes were underneath the chair.

Here "under*neath*" might additionally imply that they were also out of sight -- both below, and covered-up.

Note that there are almost no cases here were switching the terms would be ungrammatical. Between these two intense -neath terms there are some uses that are by rote, and there is no logic. Different kind of hiding and different kinds of secrets may use each word:

- Something hidden far under the ground is "deep beneath the earth".
- Someone hiding true feelings may really care "underneath it all".

So, where should your hidden bunker be? Either works well.