Backpropagation, which most researchers will agree is an AI algorithm, is a "simple algorithm".
So are many other AI algorithms, some of which are simple enough to be understood so well that most people don't recognise them as AI anymore: search algorithms like depth- breadth- or best-first search, game-playing algorithms like alpha-beta minimax, gradient descent/ hill climb, are the examples that readily come to mind.
I think the above article and your comment are assuming that, for an algorithm to be "AI" it must be very complicated and difficult to understand. This is common enough to have a name: "the AI effect". A few years down the line I bet people will say that "this is not AI, it's just deep learning".
There's no reason for AI algorithms to be complicated. Very simple algorithms can create enormous complexity, even infinite complexity. The state of deterministic systems with even a couple of parameters can become impossible to predict after a small number of steps if they have the chaos property. Language seems to be the application of a finite set of rules on a finite vocabulary to produce an infinite set of utterances. Complexity arises from very simple sources, in nature.