How to get an advisor for my amateur research? I'm about to embark on a research project in complex systems on my own spare time. It seems like the idea I have is a good one. However, it would be better, for obvious reasons, to have an experienced person guide me. What are the chances that a university professor would agree to guide me while I try and churn this out? (Obviously, I want it to be as professional as possible). 

If I don't get anyone on board, what are my chances of publishing, or at least getting a pat on the back from admissions committees when I apply for graduate school? Do they appreciate this kind of thing?
You've mentioned that you intend to apply to grad school.  It would be helpful to know where you are in the academic process: high school, started university, finished university?
Regarding asking for help, I see two distinct phases:

If you have not yet started the project (and you are not currently a student somewhere), then I think most professors and professional researchers would be very wary to spend any time helping you.  Complex systems, is, in particular, a field that attracts a lot of crackpots, and even the most well-intentioned projects of interested amateurs usually fizzle.  I think the best bet would be to find someone you already know to provide early consultation, rather than cold-calling a professional at this stage.
If you are enrolled in a university of some kind, then the story is different: the faculty has something of an obligation to talk to eager students. (-:

If you have already made significant progress and have specific technical questions, I think almost anyone would be wiling to help.


One other piece of advice: don't worry too much about talking about your idea.  Many 'amateurs' get obsessed with the secrecy of their amazing idea -- this is the road to crackpotdom.  I think it would be quite reasonable to state your idea on a site such as mathoverflow and ask whether it is a credible research proposal.
If you can cite recent publications in refereed journals that indicate active interest in the area of your proposed work, then this is a good sign that you have both done your homework and identified an interesting subject of investigation.
That said, even if the thing you are investigating turns out to be interesting only to you, it could still be worth continuing.  It might lead somewhere more interesting later, and you would certainly develop useful skills in the process.

If I don't get anyone on board, what are my chances of publishing, or at least getting a pat on the back from admissions committees when I apply for graduate school? Do they appreciate this kind of thing?

A successful independent project, especially one resulting in a publication, would be great material for your grad school app.   If your project does have interesting results, it should not be too hard to publish, regardless of your credentials.
Even if it turns out that your work is not publishable in a refereed journal, it might find a home in your school's &quot;journal of undergraduate research&quot; (or something similar), which would also be a nice resume bullet during the grad school application procedure.
Finally, it's probably best to simply avoid the word &quot;amateur&quot;.