Not sure about unique but The Bloomsbury Club is excellent.

You should also check out the Experimental Cocktail Club in Chinatown.

If you’re after weird/science-like cocktails, The Alchemist is fun.

I thought that the Mr Fogg’s series of cocktail bars was going to be cheesy but they are extremely well executed and the cocktails are superb - definitely worth a visit.

My personal favourite cocktail bar in London is Freud on Shaftesbury Avenue

My favourites are Happiness Forgets and Found Bar both in shoreditch. friendly and good value speakeasy type places

If it’s a “classic” cocktail experience you’re after then some of the high end hotels have good spots. The Connaught Bar and the Library Bar at the Lanesborough are very nice (quite expensive though and a bit stuffy).

Personally I’m a fan of Bar Américain at Brasserie Zédel.

Cahoots is another decent cocktail bar with a London Underground theme (set in an old station).

Not a cocktail bar but the Southbank Centre has a cool rooftop garden area with a small bar.

Nightjar is worth a mention. Excellent cocktails with live jazz. Reservations basically essential.

The Cocktail Trading Co, on the north end of Brick Lane is what you’re after. This is an amazing cocktail bar that’s small, charming, has a seasonal custom menu, can make all the classics with ease, and has the friendliness staff.

Went to Roof East in Stratford the other day, unpromising location on top of a multi story car park but nice bar and looks like it had some interesting activities!

I don’t know about their cocktails, but Bar Elba in Waterloo is a nice rooftop bar – good vibes for a medium-to-large social event. The Alchemist (like Scott suggested in another comment) has very interesting weird cocktails, but personally, the novelty wore off after a couple of visits as the cocktails themselves are… OK.

Other nice cocktails bars that I can recommend are Callooh Callay and Cahoots.