No camera does well with fluorescent lights. The shutter or image capture process can sometimes catch a light in between color shifts.
All fully automatic cameras will have a problem with fluorescent lighting. It depends on the shutter speed and shutter type. The colors are shifting at 120 times per second. A high speed shutter or different lighting is the only answer. Open the windows, use incandescent lighting or even try the flash (not very effective).
The iPhone 5 and 6 will have the same problem. Even a professional DSLR will have trouble under those conditions.
Actually my old iPhone original didn't have this problem, nor did my friend's iPhone 3G. Nor does my panasonic lx4 (nor of course my Nikon d300). My iPhone 4 is truly terrible in this regard.
It would seem that the fix would be easy: give the user a few pre-set white balance settings to choose from, for extreme circumstances.