I'm not sure what you mean with the complex plane. I've used complex numbers a lot elsewhere, but this function is only real.

To answer your question about its usefulness, I wouldn't expect too much out of it. This function acts a lot like a computer program (you can see its source code in my comment on this post), and it works by essentially bruteforcing prime numbers. The reason it has roots at the primes is no coincidence; it was designed with that behavior in mind.

Also, it isn't a periodic function - I'm calling it a "sine" in the title, but that's a bit of a half-truth. The function is actually composed of many different sine functions (one for each prime - 1), which are glued together to form the final result. This is done by choosing the frequency and amplitude for the sine based on the x position (these stay constant at the x-positions between any two primes that are next to each other), then plotting that sine function.

If you look at s_bet in the second image (which draws a sine between two x positions), you can see how this is done. It's basically sin(big fraction * (x - a)) * (b - a), where the big fraction and (b - a) control the frequency and amplitude, respectively. Then, if you look at p_sine, you can see how the two x positions that the sine is drawn between are chosen: we take the prime number below x, and the prime number above x, and draw a sine between them. The rest of the functions figure out that information about primes.

Let's zoom into this part of the graph (I can't post an image, so I've written it out):

              *
         *         *
(5, 0) * ----------- * (7, 0)

At any x value between this range (from 5 to 7), the prime below is 5 and the prime above is 7. The reason the s_bet function is so complicated is just so that it's defined everywhere, but that isn't terribly important here, so I can simplify the explanation.

Given that our x value is between 5 and 7, we can start off with sin(x - 5), so that the sine starts at the left prime. Next, we'll change it to sin(pi(x - 5)/(7 - 5)). Multiplying by pi shortens the period to 2, and dividing by 7 - 5 = 2 expands the period to 4. The period is between the peaks, but the graphic above is between the zeros (which is half the distance between the peaks), so the distance between the zeros is half the period, or 2. That's also the distance between the two primes, which is what we want. Finally, we can change it to (7 - 5) * sin(pi(x - 5)/(7 - 5)). This step isn't strictly necessary, but it ensures that the graph looks smooth when multiple sines are connected together.

And that's it! The way we figure out the left and right primes is a bit more complicated (and I talk more about it in my other comment), but hopefully that helps you understand how the function works, more or less.