Because you have never heard of a concern with swimming and bathing in chlorinated pools... research has been ongoing for many years as to the long term and short term health affects of chlorine exposure. I am not one to say you should not swim or shower, but I will definitely read and understand the studies. Alternatives to chlorine have been widely used and the health affects of these alternative treatments are also being studied. Copper, silver, bromine, ozone, UV, even KDF have all been widely used and all have their merits. Chlorine is cheap, effective, and important. Without it, pools would be nothing mnore than a dirty communal bath that is a fecal cesspool. Municipal and well supplies that have the potential for certain types of bacteria should be chlorinated, then dechlorinated prior to consummption and use.
Not recommended, but I have done it. Spray bleach into a shower to clean it (dont get it on the chrome), come back in three minutes and see how long you can stay in the bathroom without your eyes and lungs burning. Lower levels are not enough to cause this reaction, but a shower is the perfect place to atomize and maximize what little chlorine is in the water supply, maximizing your exposure to it.
Removing chlorine is cheap, simple, and highly recommended.