How to add extra field in admin comment section

You can add it directly through your functions.php, unfortunately this is not documented very well at all.

You want to hook into add_comment_meta using add_action, to look at this further refer to line 500-ish of comment.php.

It looks like this

function add_comment_meta($comment_id, $meta_key, $meta_value, $unique = false) {

    return add_metadata('comment', $comment_id, $meta_key, $meta_value, $unique);

}