using 4.6.0
Let me know if you want any more info, not sure what would be helpful

would expect to see this - http://elixir-lang.org/docs/stable/elixir/Application.html
If you mean the Application from the standard library, then yeah, that makes sense, completion only works for parsable-code and your standard-library is only available in precompiled .beam-files.
This may actually be fairly simple because the rules for Elixir modules: just scan the paths for Elixir.*.beam files and * is the name of the aliases that can be completed. This may be a feature that can be added quicker than completion for functions in .beam-only modules.
Would it be possible to utilize elixirs autcompletion capabilities, since it's already installed on the system?
A project doing this that way is https://github.com/wende/autocomplete-elixir - take a look at https://github.com/wende/autocomplete-elixir/blob/master/lib/alchemide/autocompleter/autocomplete.exs
Edit: A better implementation: https://github.com/tonini/alchemist-server
Works after #574

great work man, excited for 4.8 :D I think when this lands, we'll have the nicest completion out of all the various implementations, which is quite an accomplishment!
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Works after #574