A root guide showed up online for the Galaxy S4 already, before the phone is even physically released. Samsung Galaxy phones are fairly easy to root and this is not the first time that a device was rooted before its release. Although unlikely, Samsung may block this root method with a patch by changing its bootloaders and kernel.
Developers at XDA Developers claim that a root is available for the Samsung Galaxy S4 and there's even a video guide to show the method. Unfortunately, the root process is said to work for the Exynos 5 Octa variant of the Galaxy S4, model GT-I9500. Most of the devices on the market will have the Snapdragon 600 processor, so this leaves out a portion of potential Galaxy S4 owners.
Chainfire, a Samsung developer that specializes in root, said that Samsung may have added protection on the Galaxy S4. He also said that this would be a great accomplishment if the root actually works, since there are not many Samsung Galaxy S4 devices out now. There's a possibility the developers may have jumped the gun a bit: for many past devices a modified stock ROM was flashed to the device to achieve root access. He explains that there is more to it than just incorporating files into the ROM and that he tested the standard method on the Galaxy S4 variant GT-I9505. The process failed, due to new protection on the Galaxy S4 that prevented the phone from being rooted.
Chainfire said on Google+: "I would like to point out that we've tried this trick with the I9505 weeks ago (and we actually tested it on a real device), and this time around, it is not a simple case of injecting the files. There is some further protection in the firmware that prevents the su binary (which is needed for root access) from functioning properly — without any known workaround at this time."
