When I'm looking for a partner, I want them to share my personal values or at least hold very similar ones. I also want them to be in sort of the same state of mind as I am - such that they have personal goals that are in line with my personal goals, and that we have enough shared experiences to understand where one another is coming from. I think these things are important to have in common because they minimize the possibility of conflict over things that are integral to who each person is.
I also want someone who sort of makes up for my shortcomings, however - I'm naturally a very shy, withdrawn sort of person, and I enjoy having people around who are friendly and spontaneous because they usually know how to draw me out of my shell.
My knee-jerk response is to look for similarity so we have something in common to start bonding over... after that, if we open up and find that we have some differences too, that's great too. It creates some variety. I don't want someone who's exactly like me, but I also don't want someone who I can't relate to at all. I want something in between. However, many social psychology studies show that individuals are more likely to be attracted to and form relationships with those who are more similar to them for many reasons (I won't get into it now).
The truth is that I'm still learning and thinking about what things I want to be similar and different between me and a partner, but similar values and life goals should be somewhat similar.
More similar is ideal. At least where things matter. IF there's any difference where you both consider something important, it will almost never work. Some core hobbies should line up, obviously, and if you both find having a kid or certain careers or whatnot important, it matters you two line up. Or if one or both think the religious affiliation matters, that's something that needs to match up. But ya, there should be a FEW differences so it's not stale, though, honestly, even if there were no differences, as long as both of you are comfortable/happy with how things are and/or actually try new things together, there would be no problem.
Honestly, people who think "opposites are ideal" are fools that don't realize that'll definitely end. The friction differences (and more importantly the stubbornness of humans to tolerate and accept differences) cause are why we have all the conflicts in the world in the first place... don't be naive to think differences will not do the same in regards to love/relationships. In fact, I'd say they'd have an even greater friction/problem on such.