It would extend north to a new station I've called Colindale South, which would be served by this, Thameslink, and the Northern line. The idea is to decongest the Northern line, since by Hampstead the Edgware branch of the Northern line is insanely busy at peak hours.
The soutbound extension would require a simple new junction to be built to allow the trains to go from Whitton to Strawberry Hill, as well as 2 new platforms with turnback capabilities at Teddington as seen in picture 2. The advantage here is that they can use (and slightly expand, there's space for it) Strawberry Hill depot while having a terminus that's much easier to work with. Hounslow (which I've renamed to Hounslow Heath to avoid confusion) doesn't have turnback infrastructure in place and would have to turn trains around, which is messy and inefficient.
The Charles line btw is Crossrail 2, this is just for future reference if/when it gets built (and assuming the plans for it to go to Shepperton via Teddington still happen).