Independent search box for views taxonomy page
You can achieve this in Drupal 7 and 8 using a mixture of modules such as:

Search API


  This module provides a framework for easily creating searches on any
  entity known to Drupal, using any kind of search engine. For site
  administrators, it is a great alternative to other search solutions,
  since it already incorporates facetting support and the ability to use
  the Views module for displaying search results, filters, etc. Also,
  with the Apache Solr integration, a high-performance search engine is
  available for this module.


Then you can use: Search API Autocomplete to accomplish the auto complete functionality you require


  The Search API Autocomplete module in action. This module provides
  autocomplete functionality for Search API searches.


And all of this is managed and is customizable via Views module and having a Search API Server that indexes what you want (Here are taxonomy terms for your case)

If you want to make it easier setting things up you can try Search API Pages


  This module is for creating simple search pages for the Search
  API project, not using Views or any other modules. They can be used
  when a view would be unnecessarily slow or complex, or for quickly
  testing out functionality. They also provide search blocks for
  starting a search from anywhere on the site.