Wrap jQuery AJAX in ES6 Promise

jquery-ajax-promise.js

function AjaxError(jqXHR, textStatus, errorThrown) {
  this.name = "AjaxError";
  this.message = textStatus;
  this.jqXHR = jqXHR;
  this.errorThrown = errorThrown;
}
AjaxError.prototype = new Error();
AjaxError.prototype.constructor = AjaxError;

(function($) {
  function decorateAsJQuery(promise) {
    promise.done = function(fn) {
      return decorateAsJQuery(promise.then(fn));
    };
    promise.fail = function(fn) {
      return decorateAsJQuery(promise.then(null, fn));
    };
    promise.complete = function(fn) {
      return decorateAsJQuery(promise.then(fn, fn));
    };
    return promise;
  }
  var jqAjax = $.ajax;
  $.ajax = function ajax() {
    var args = Array.prototype.slice.call(arguments);
    var jqPromise = jqAjax.apply(this, args);
    var promise = new Promise(function(resolve, reject) {
      jqPromise.then(function(data, textStatus, jqXHR) {
        resolve(data);
      }, function(jqXHR, textStatus, errorThrown) {
        reject(new AjaxError(jqXHR, textStatus, errorThrown));
      });
    });
    return decorateAsJQuery(promise);
  };
})(jQuery);

What are the advantages of doing this over using the simple http://api.jquery.com/category/deferred-object/?

To the best of my knowledge, jQuery's deferred object aren't Promise/A+ compliant. In particular, they don't catch errors in reactions.
You can find numerous blog posts on the subject that will explain the the subtle differences. For instance, I suggest you have a look at JavaScript Promises, by Jake Archibald

jqXHR is a superset of the browser's native XMLHttpRequest object.

Such as: abort current request

var jqXHR = $.ajax(...)
jqXHR.abort()

Is this free software and if yes, would you mind explicitly putting a license on top? I would very much like to reuse it in a project.

This breaks the jQuery AJAX load-function ($(element).load(someURL)), since jQuery 2.2.0 and 3.0.0 (see jquery/jquery@97ef1f2), as the .always() function is used instead of the deprecated .complete().

This can be fixed by adding the line promise.always = promise.complete; between lines 20 and 21.

Also, the reason why you would use this is to be able to use .catch() on a chain of promises, so that you can have a single error handler if any one call in a series of AJAX calls fail.