The idea, which has existed since at least TfL's Transport 2025 document (released 2006), is to have two, distinct lines: one running from Edgware to Kennington (and possibly beyond to Battersea) via Charing Cross, the other from High Barnet to Morden via Bank. The current cross-dressing at Camden Town, where the lines flip over each other, would come to an end, meaning that if, say, you were travelling from Archway to Embankment, you would need to change at Camden. Same goes if your trip runs from Hampstead to Old Street.
Responding to questions by a Lib Dem parliamentary candidate, Northern line general manager Pat Hansberry confirmed that TfL plans to split the line for the evening rush hour by the end of the decade, and said that the system was already being implemented for northbound trains in the morning.
Advantages? An increase in trains on the two lines (from 22-25 trains per hour up to 30 on the central London sections), meaning an increase in capacity. Disadvantages? Well, there'd be an increase in the number of passengers switching trains at Camden Town, something that may require the station to be partly re-built. We've witnessed recently at Edgware Road the chaos that can result from ill-thought through changes to Tube line behaviour.
It's possible (though unlikely) that they may give the semi-separate lines distinct colours on the Tube map, and even different names. In the interest of geographical equality, we'd quite like to see the branch down to Morden re-christened the Southern line.