No, pool water does not turn purple when you pee in it, but that does not mean that you can just go ahead and do it. This video explains which chemical processes occur when urine ends up in the pool water.
Stagnant water, such as that in swimming pools are ideal breeding grounds for dangerous microorganisms, such as salmonella, E. Coli and giardia. That's why swimming pools need to be disinfected, which is usually done with chloride, sodium hypochlorite and calcium hypochlorite.
These substances react with the water, but also with the swimmers, their sweat, dirt and urine. Because urine contains ureum, a waste product of protein metabolism. When ureum reacts with chloride, it turns into nitrogen chloride, the substance which causes the typical stinging smell in swimming pools. Irritated eyes and skin after swimming can also be a result of contact with nitrogen chloride. But what's worse: substance is also linked to respiration difficulties, such as asthma.
According to the American Chemical Society especially that last affliction is a problem for professional swimmers who spend many hours in the water every day. However, the scientists want to give a small warning and urge swimmers to take a shower before swimming and leave the pool to go pee, since that is what toilets are for.