This depends on your perspective, I think. Any game mechanic is complex if you're not familiar with it. There are a lot of Pokemon types, each with their own unique weakness/resistance, and a LOT of Pokemon, and if you don't have a head for remembering these things, then it's an absolute minefield.
Looking at it from a vaguely neutral perspective and in comparison to other video games that utilise an elemental system - and god, there are a LOT of these; it's practically an RPG staple - I think that dual-types are generally not complex in how they are used in game, but are of significantly greater depth than the norm. Most RPGs will give something a single element, and usually there are no more than four or five elements. Pokemon has eighteen. With dual properties. That's greater variety, but does variety equate to complexity? Personally I don't think it does. A complex game mechanic, to me, is one that is difficult to manipulate and utilise even after you know how it works, and type matchups are very simple and easy to understand, once you know what you're doing. Once you know what is weak to and resists what, that's pretty much all there is to it. It's just getting to that point which may prove troublesome for some.
So, no. Dual-types are an expansion of a simple game mechanic that is just expanded upon significantly in Pokemon when compared to other games, would be my answer.
