There already is a better Turing test, which is the Turing test as originally described.
To run the test as originally described, you need an active control; a human conversing with the judges at the same time in the same manner, where their decision is "Which is the human?", not "Is this a human?" If the incompetent judges had been also talking simultaneously with a real 13-year-old from Ukraine, I have no doubt that Eugene Goostman would have bombed horribly.
This is not that much better. The article that The Most Human Human is based on talks about the difficulty of communication in a 5-minute window, and the lack of knowledge lay judges have about what AI involves. The author consistently got named a human by better-applying the tactics of the some of the most successful bots: controlling conversation flow and using humor.
It's an improvement, but a winner would still win by "gaming" judges' psychology.
So the Turing test has been "passed", and the general consensus is that this was achieved in a very unimpressive way - the 13 year old Ukrainian persona was a cheat, the judges were incompetent, etc... These are all true, though the test did pass Turing's original criteria - and there are far more people willing to be dismissive of those criteria in retrospect than were in advance. It happened about 14 years later than Turing had been anticipating, which makes it quite a good prediction for 1950 (in my personal view, Turing made two mistakes that compensated - the "average interrogator" was a much lower bar than he thought, but progress on the subject would be much slower than he thought).
But anyway, the main goal now, as suggested by Toby Ord and others, is to design a better Turing test, something that can give AI designers something to aim at, and that would be a meaningful test of abilities. The aim is to ensure that if a program passes these new tests, we won't be dismissive of how it was achieved.
Here are a few suggestions I've heard about or thought about recently; can people suggest more and better ideas?
My current method would be the lazy one of simply typing this, then waiting, arms folded:
"If you want to prove you're human, simply do nothing for 4 minutes, then re-type this sentence I've just written here, skipping one word out of 2".