About the complex nature of the wave function?
More physically than a lot of the other answers here (a lot of which amount to "the formalism of quantum mechanics has complex numbers, so quantum mechanics should have complex numbers), you can account for the complex nature of the wave function by writing it as $\Psi (x) = |\Psi (x)|e^{i \phi (x)}$, where $i\phi$ is a complex phase factor. It turns out that this phase factor is not directly measurable, but has many measurable consequences, such as the double slit experiment and the Aharonov-Bohm effect.
Why are complex numbers essential for explaining these things? Because you need a representation that both doesn't induce nonphysical time and space dependencies in the magnitude of $|\Psi (x)|^{2}$ (like multiplying by real phases would), AND that DOES allow for interference effects like those cited above. The most natural way of doing this is to multiply the wave amplitude by a complex phase.
This year-old question popped up unexpectedly when I signed in, and it's an interesting one. So I guess it's OK just to add an intuition-level "addendum answer" to the excellent and far more complete responses provided long ago.
Your kernel question seems to be this: "Why is the wave function complex?"
My intentionally informal answer is this:
Because by experimental observation, the quantum behavior of a particle far more closely resembles that of a rotating rope (e.g. a skip rope) than it does a rope that only moves up and down.
If each point in a rope marks out a circle as it moves, then a very natural and economical way to represent each point along the length of the rope is as a complex magnitude. You certainly don't have to do it that way, of course. In fact, using polar coordinates would probably be a bit more straightforward.
However, the nifty thing about complex numbers is that they provide a simple and computationally efficient way to represent just such a polar coordinate system. You can get into the gory details mathematical details of why, but suffice it to say that when early physicists started using complex numbers for just that purpose, their benefits continued even as the problems became far more complex. In quantum mechanics, their benefits became so overwhelming that complex numbers started being accepted pretty much as the "reality" of how to represent such mathematics.
That conceptual merging of complex quantities with actual physics can throw off your intuitions a bit. For example, if you look at moving skip rope there is no distinction between the "real" and "imaginary" axes in the actual rotations of each point in the rope. The same is true for quantum representations: It's the phase and amplitude that counts, with other distinctions between the axes of the phase plane being a result of how you use those phases within more complicated mathematical constructions.
So, if quantum wave functions behaved only like ropes moving up and down along a single axis, we'd use real functions to represent them. But they don't. Since they instead are more like those skip ropes, it's a lot easier to represent each point along the rope with two values, one "real" and one "imaginary" (and neither in real XYZ space) for its value.
Finally, why do I claim that a single quantum particle has a wave function that resembles that of a skip rope in motion? The classic example is the particle-in-a-box problem, where a single particle bounces back-and-forth between the two X axis ends of the box. Such a particle forms one, two, three, or more regions (or anti-nodes) in which the particle is more likely to be found.
If you borrow Y and Z (perpendicular to the length of the box) to represent the real and imaginary amplitudes of the particle wave function at each point along X, it's interesting to see what you get. It looks exactly like a skip-rope in action, one in which the regions where the electron is most likely to be found correspond one-for-one to the one, two, three, or more loops of the moving skip rope. (Fancy skip-ropers know all about higher numbers of loops.)
The analogy doesn't stop there. The volume enclosed by all the loops, normalized to 1, tells you exactly what the odds are on finding the electron along any one section along the box in the X axis. Tunneling is represented by the electron appearing on both sides of the unmoving nodes of the rope, those nodes being regions where there is no chance of finding the electron. The continuity of the rope from point to point captures a rough approximation of the differential equations that assign high energy costs to sharp bends in the rope. The absolute rotation speed of the rope represents the total mass-energy of the electron, or at least can be used that way.
Finally, and a bit more complicated, you can break those simple loops down into other wave components by using the Fourier transform. Any simple look can also be viewed as two helical waves (like whipping a hose around to free it) going in opposite directions. These two components represent the idea that a single-loop wave function actually includes helical representations of the same electron going in opposite directions, at the same time. "At the same time" is highly characteristic of quantum function in general, since such functions always contain multiple "versions" of the location and motions of the single particle that they represent. That is really what a wave function is, in fact: A summation of the simple waves that represent every likely location and momentum situation that the particle could be in.
Full quantum mechanics is far more complex than that, of course. You must work in three spatial dimensions, for one thing, and you have to deal with composite probabilities of many particles interacting. That drives you into the use of more abstract concepts such as Hilbert spaces.
But with regards to the question of "why complex instead of real?", the simple example of the similarity of quantum functions to rotating ropes still holds: All of these more complicated cases are complex because, at their heart, every point within them behaves as though it is rotating in an abstract space, in a way that keeps it synchronized with points in immediately neighboring points in space.
Alternative discussion by Scott Aaronson: http://www.scottaaronson.com/democritus/lec9.html
From the probability interpretation postulate, we conclude that the time evolution operator $\hat{U}(t)$ must be unitary in order to keep the total probability to be 1 all the time. Note that the wavefunction is not necessarily complex yet.
From the website: "Why did God go with the complex numbers and not the real numbers? Answer: Well, if you want every unitary operation to have a square root, then you have to go to the complex numbers... " $\hat{U}(t)$ must be complex if we still want a continuous transformation. This implies a complex wavefunction.
Hence the operator should be: $\hat{U}(t) = e^{i\hat{K}t}$ for hermitian $\hat{K}$ in order to preserve the norm of the wavefunction.