This really depends on how complex you're interested in getting. But here's a bunch of games I love, that are great for two players, in order of least to most complex (roughly):
Ticket to Ride: Best "gateway" game there is. Simple rules, basically just a "set collecting" game. If you have an iOS device, the app is a great way to try it out.
Carcasonne:This is my favorite game to play with my girlfriend. The rules couldn't be simpler (draw a tile, play a tile), but there is an incredible depth to the strategy, and it has a ton of expansions to check out. This game also has a great iOS app.
Small World: Think Risk, but only 45 minutes long. Rules are dirt simple, is super replayable, and scales really well with more players. Again, the iPad app for this game is great.
Lords of Waterdeep: I actually don't love this game, but I know enough people who do that I feel ok recommending it. It's what you'd call a worker-placement game. And hey, guess what, check out the very well done iOS app.
Pandemic:This is the best co-op game I know, and it's great for two players. The basic premise is you're a team of scientists traveling the world, saving the planet from a flurry of terrible diseases. It's not hard to get your head around the rules, but it's exceptionally difficult.
Hive:I've seen this described as bug-themed chess, and that's pretty accurate. This game also comes in an awesome portable version, that's fairly cheap. (This game has an app too, but it's uh... not great.)
Memoir '44: This is a war game themed around WWII, and it's great in its simplicity. I know you said in your post that you want something "deep/complex," but honestly, if you aren't neck-deep into tabletop games, most war games are so complex that they tend to scare people off of the hobby. That being said, I've been playing tabletop games for years, and this is still my favorite war game. Check out some "how to play" videos.
Race for the Galaxy: This game is a beast, so I won't really try and explain it. there's been a ton written online about it. I do love it, though, and it's a fantastic puzzle to try and figure out. And if you like, there is a near infinite number of expansions.
Other people have mentioned X-Wing and Netrunner, both of which are games that I love, but can't really afford to dive into.
I would also like to mention Sherlock Holmes: Consulting Detective which is supposed to be an incredible experience with two players, but I haven't been able to play it since it's been out of stock everywhere. Supposedly, they're shipping another batch sometime this Spring.