No. Not that kind of protein complementing!
Back in 1971 there was a very popular book called "Diet for a Small Planet". In this book was put forth the idea that if you were a vegetarian you needed to mix types of protein to get a compete set of the 4 amino acids that are not made by the body.
So for example rice and beans went well together because each was high in a different 2 of the 4 amino acids...both together gave you all 4 in good amounts.
Even though I was not a vegan, I have leanings in that direction and so I always remembered that part of the book. (as well as the parts about how terribly inefficient it is to grow meat).
These days I'm a vegetarian about 85% of the time (I eat dairy, but only skim milk)...the rest of the time I'll eat meat or fish as along as it's damn good stuff.
Recently, I was eating lunch with a co-worker (a pretty solid vegan) and he didn't know about this at all (!). The reason is: when we got back to our offices he looked this up and it TURNS OUT IT IS NOT TRUE.
Here's part of the wikipedia entry for Diet for a Small Planet;
Knowing that her audience would be skeptical that a vegetarian diet could supply sufficient protein, much of the book is devoted to introducing her theory of complementing proteins, also called protein combining. This is a method of eating different plant foods together so that their combined amino acid pattern matches that of animal foods. But while Lappé was correct that combining would indeed result in a more meat-like protein profile, it is also unnecessary: Individual plant foods contain all the amino acids required by humans, in amounts which satisfy growth and maintenance; however, certain deficiencies of particular amino acids should be considered since such deficiencies can have a negative effect on health. In other words, mimicking the composition of animal proteins is not essential to human nutrition. After this was pointed out, Lappé recanted the idea of protein combining in the 10th anniversary 1981 version of the book:
- "In 1971 I stressed protein complementarity because I assumed that the only way to get enough protein ... was to create a protein as usable by the body as animal protein. In combating the myth that meat is the only way to get high-quality protein, I reinforced another myth. I gave the impression that in order to get enough protein without meat, considerable care was needed in choosing foods. Actually, it is much easier than I thought.
- "With three important exceptions, there is little danger of protein deficiency in a plant food diet. The exceptions are diets very heavily dependent on  fruit or on  some tubers, such as sweet potatoes orcassava, or on  junk food (refined flours, sugars, and fat). Fortunately, relatively few people in the world try to survive on diets in which these foods are virtually the sole source of calories. In all other diets, if people are getting enough calories, they are virtually certain of getting enough protein."
(Red parts emphasized by me)
So there you go. It's even easier to eat vegan than I thought.