Is this on a local server or a remote/hosted one?
Fair enough. Just wanted to check as running WP_DEBUG on a publically accessible server isn’t advisable.
This issue isn’t going to fix itself, so turning debug off won’t really help. The likely suspects are your theme and your active plugins. Does the warning disappear when using the default Twenty Thirteen theme with all plugins deactivated?
This is a known problem with XAMPP Version 1.8.3 and any other hosting that uses PHP Version 5.5 (or newer).
There is more info and some links here:
http://wordpress.org/support/topic/wordpress-core-and-mysqli_connect?replies=2
I just ran into this problem last night and switched to XAMPP 1.8.2 which is fully supported by the XAMPP folks, too.
You can use the following library if you do not have mysql_connect enabled by have PDO enabled:
https://github.com/AzizSaleh/mysql
Copy those files to a folder and then in your wordpress wp-includes/load.php file replace the following two lines:
wp_load_translations_early();
die( __( 'Your PHP installation appears to be missing the MySQL extension which is required by WordPress.' ) );
With the following:
$pathToMySQL = 'C:\\xampp\\htdocs\\mysql\\';
require_once($pathToMySQL . 'MySQL_Definitions.php');
require_once($pathToMySQL . 'MySQL.php');
require_once($pathToMySQL . 'MySQL_Functions.php');
