A GOOD (not complex) story will in my opinion always improve a game, but it will never fully make up for bad game play.
As for a COMPLEX story, I would usually disagree. A story doesn't have to be long and complicated to be good.
I give the example of the Half-Life series. First and foremost, the game play was amazing, so that's already most of the boxes ticked. But the story was itself brilliantly written.
When you think about it, not much happens (SPOILERS coming up, but if you haven't played it yet then I have nothing but pity for you). Gorden puts on a cool suit, fiddles around with an experiment, everything goes wrong, and he has to fight his way out Black Mesa. We learn about Xen, and the G-Man.
Then in HL2 (and I won't go into the episodes yet), Gorden wakes up on a train, learns (pretty quickly) that the Combine have invaded, and goes about saving the day again by destroying the citadel.
On the face of it there isn't really that much of a story. There aren't hundreds of twists and turns, and those that do exist fall much more into the category of level design, rather than story telling.
But what makes the story good is all the subtle things. The way Gorden always catches glimpses of the wider world around him. The sense of mystery and intrigue that is built layer by layer. All of this could be completely ignored by some players, and they'd enjoy a great FPS. But for those that do notice it, it's a blessing: without being bamboozled by hundreds of characters, tens of factions and journeying across the planet, it makes a great game amazing.
OK, Half-Life fanboyism over. The point is a well crafted story can only be a good thing for a game. BUT, to directly answer the post, a COMPLEX story can sometimes even make a game worse. So I voted no.