Hi Charly,
For the live search it is not possible unfortunately. It was indeed a planned feature for a long time, but due to the variety of themes and the lack of consistent development guidelines on them, it’s still not possible to implement this feature to work on most of them.
It is however possible to enable a search override feature, which allows displaying the search results found by the plugin on the results page: https://wpdreams.gitbooks.io/ajax-search-pro-documentation/content/behavior/results-page-override.html