jeremypreed

Improved Search is a AJAX search interface you can place anywhere on your WordPress site and get instant search results as soon as you beginning typing. Improved Search delivers the most relevant results first. It also separates results by post type to help you quickly identify the post you are trying to find. Improved search allows you to customize the experience to fit the needs of your WordPress site. Improved Search comes with a shortcode as well as a widget for you to control its location.

Improved Quick Search

Improved Search is a AJAX search interface you can place anywhere on your site and get instant search results as soon as you beginning typing.

wokamoto

You can choose (child) categories and see the posts belong to the chosen category with AJAX.
When the category you choose from the list box has child categories, the child categories will be shown in a new list box. 
Clicking the “Search” button, the list of posts belong to the bottom category which you choose will be displayed under the “Search” button.

Simple Category Search

You can choose (child) categories and see the posts belong to the chosen category with AJAX.

Improved Quick Search vs Simple Category Search