Well personally i think your pc would play it very well? high-ultra high.
if it was me then no i wouldn't buy a new gpu, especially just for one game!
you have a very nice pc
One £30 game isn't worth £hundreds in upgrades!
First of his question was should he get a new monitor or a new video card ...... Secondly a 6870 is NOT running the game bf3 on high/ultra settings.
It all depends I find gaming on a 23-30 inch monitor at 1920x1080 with graphics cranked my sweetspot, Im talking depth of field/tessalation 8X/16X etc. Its all about whats right for you, I cannot stand the Bezels in multi monitor setups. So my next upgrade will be a 30+ with higher res. Monitor could be a good upgrade/investment. As your monitor will not become obsolete as long as its one you want and will use for years to come ......My wifes 24 inch I have had for 7 years and it still looks awesome.
Like stated about for max settings you may want to get a new video card, as recommended is a 560Ti or 6950 which in the VGA hierarchy charts is a few steps above your current card and those two cards cannot max the game on Ultra. Med settings I think is your current sweet spot. Hope this helps.