Idea Summary
Now it is only possible to share apex collections in the same application, but if you try to read its data from another application you can't do it.

Use Case
In very big projects, applications are treated as modules but they can share some logics, and appex collections would be a very useful way to share info between them temporarily.

Preferred Solution (Optional)
Allow to create apex collections at session level instead of application level. By default, if you don't specify it it would create the collection for the current application. But with some optional additional parameter, you should be able to make it available to all apps.