Do I need neocomplete in addition to vim-elixir and alchemist.vim?
Does the vim-elixir/alchemist.vim combination also provide that on their own?
No and no. To use vim-elixir and alchemist.vim you do not need, but for completion do you need of something extra. In the Alchemist.vim docs shows that you can use deoplete or YouCompleteMe for completion (I do not know if works with the neocomplete).