I was interested in doing some research in complex analysis. I already have a basic understanding of the subject. i.e. I read Saff and Snider's book "Fundamentals of Complex Analysis". But now I would like to write a paper in my free time for fun. But I have no idea whatsoever what a beginner in complex analysis could do a paper on. So I was wondering if anyone had any suggestions as to a topic that I could research more in depth and write a paper on. Thanks :)
If you know how to program, a very fun project would be to render different kinds of fractals. The most famous, of course, is the Mandelbrot fractal, defined as the set of points $c$ in the complex plane so that if $f(z) = z^2 + c$ then the sequence $f(0), f(f(0)), f(f(f(0))), \ldots$ is bounded. It's not hard to render an image of the fractal: for each pixel on the image, take the corresponding point $c$ and color it black, if after (say) $100$ steps the sequence is still within two units of the origin. If not, you can color it various shades based on how how many steps it took to leave this disc.
You've probably seen pictures of the Mandelbrot set, but what's surprising is how simple the algorithm is. You can make amazing images with only twenty or so lines of code. You can mix it up by changing the coloring scheme or zooming in on different parts. By taking images at successive zoom levels, you can make a nice animation. You can also try replacing $f(z) = z^2 +c$ by other functions like $f(z) = z^3 + c$ or $f(z) = e^z + c^2$ or whatever you want - most of the time, you will get a very striking, surprising image. Then you can try to explain some basic features of the picture using mathematics.
Other nice do-it-yourself fractals are the Julia sets, Buddhabrot, Newton fractals, the burning ship fractal. If you don't know how to program it from scratch, there should be software that can help you (although I haven't tried these.)
Another fun thing to try is video feedback, although it's not as directly related to complex analysis. You can rent a video camera, display the camera's footage to a screen, and then point the camera at the screen so that it records itself. If you tilt the camera at different angles you can get very interesting results. See this tutorial on Youtube.
Thanks for the clarification.
If you're up for a second, much deeper look at complex analysis, you will find that there are a great many beautiful things that can be done with the theory. My suggestion is to pick up a book that treats complex analysis rigorously and explore the topics therein. Your book already seems to address many of the applications of complex analysis (fractals, applications in celestial mechanics, etc.). Other books will address even more topics: for example, Complex Analysis by Stein and Shakarchi addresses the Riemann zeta function and the prime number theorem, both worthwhile topics for an expository paper.
If you are confident in your abilities, then there may be research topics that are accessible to you involving complex analysis: I believe there is an REU at Cornell that sometimes offers analysis on fractals as a topic, though I don't know many details about that. Perhaps a professor at your local college may be willing to advise you on the topic and tell you what you need to know to proceed in that direction.
If you want to do something that isn't really rigourous maths for a state fair how about comparing how soap bubbles form on a closed wire to the fact that holomorphic functions on a disk are determined by their behaviour on the boundary of the disk?
Shape of wire (boundary) --- determines soap bubble on the interior
Value of Holomorphic Function on boundary --- determines the value of function on interior
Useful links: http://en.wikipedia.org/wiki/Minimal_surface, http://math.berkeley.edu/~sethian/2006/Applications/MinimalSurfaces/minimal.html, http://en.wikipedia.org/wiki/Cauchys_integral_formula, http://en.wikipedia.org/wiki/Identity_theorem,