The hook you need is:
woocommerce_order_status_changed
add_action("woocommerce_order_status_changed", "my_awesome_publication_notification");
function my_awesome_publication_notification($order_id, $checkout=null) {
global $woocommerce;
$order = new WC_Order( $order_id );
if($order->status === 'completed' ) {
// Create a mailer
$mailer = $woocommerce->mailer();
$message_body = __( 'Hello world!!!' );
$message = $mailer->wrap_message(
// Message head and message body.
sprintf( __( 'Order %s received' ), $order->get_order_number() ), $message_body );
// Cliente email, email subject and message.
$mailer->send( $order->billing_email, sprintf( __( 'Order %s received' ), $order->get_order_number() ), $message );
}
}
}