If your website is in localhost or in a sub-directory (not at the root of your domain or sub-domain), the validation by ajax JQuery can work not properly. Just edit the file named ajax-check.js located in <your site>/components/com_alpharegistration/assets/scripts.

Search this first lines below :

var myDomain = '';

if ( document.domain!='localhost' ) {
    myDomain = /(http?:\/\/[^\/]+)/.exec(window.location.href)[1] + '/';
}

and replace by 

var myDomain = 'your url/';                       // example : var myDomain = 'http://www.xxxxxxx.com/xxxxx/' or var myDomain = 'http://localhost/xxxxxx/'

do not forget the last /

and comment the following lines like this

/*
if ( document.domain!='localhost' ) {
    myDomain = /(http?:\/\/[^\/]+)/.exec(window.location.href)[1] + '/';
}

*/