Need help Converting a jQuery event to a Drupal Behavior
Alter don't show because in your code: 

jQuery('.tab-pane').bind('fadeIn', function() {
             alert('fadeIn Done.');
        });


this function mean that when selecter .tab-pane fadeIn done, alert will show. But You don't call fadeIn => don't have event fadeIn. You need call that same: 

jQuery('.tab-pane').fadeIn(2000);


And better function jQuery.fn.fadeIn need outside Drupal.behaviors.fadeInEvent

(function ($) {
    var _old = $.fn.fadeIn;
    $.fn.fadeIn = function(){
        var self = this;
        _old.apply(this,arguments).promise().done(function(){
            self.trigger("fadeIn");
        });
    };

    Drupal.behaviors.fadeInEvent= {
        attach: function (context, settings) {

            $('.tab-pane').bind('fadeIn', function() {
                alert('fadeIn Done.');
            });
            $('.tab-pane').hide();
            $('.tab-pane').fadeIn(2000);
        }
    };
})(jQuery);


then you call fadeIn() every where and have event fadeIn.