There's a modern version of the argument, from Kripke in Naming and Necessity. The idea is that identity is a necessary relation, so that if x = y, then necessarily x = y. Put in the language of possible worlds, we can say that in every possible world, x exists if and only if y exists, and in every possible world where they both exist they must have exactly the same properties. In no possible world can x exist without y, and in no possible world can x have a property that y does not have.
If your mind and your body (brain) are identical, then this holds true of them. But possibly, your mind might exist without your brain (say, as a disembodied spirit, or in another body), but your brain cannot exist without your brain since this is a contradiction. So in some possible world one exists where the other does not. We can also imagine worlds where your mind has properties that your brain might lack, so that they cannot be identical.
Another way to think of it is by considering the this-world property of possibly-existing-without-your-brain. Your mind has the property of possibly-existing-without-your-brain, but your brain does not have the property of possibly-existing-without-your-brain. If they are identical, as (some types of) materialism says then this cannot be true. (See Plantinga explain it here.)
See the Modal Argument in this SEP article on dualism for more.