Your question about hidden things gets at an interesting point (even though "Under / underneath vs. etc." is much discussed on ELL). Before getting into your question specifically, I should note that frame of reference is often key to understanding these words in a wider context: "The candle was under the painting, but the map was underneath the painting" might that the candle is on a table below where the painting is hanging, while the map is behind the surface of the painting itself.
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 "underneath" 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 "underneath" 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.