Over time the outer shell of the shower filer can get tight and become hard to open. This is due to the constant changes in pressure and temperature, caused by showering.
How to open the shower filter
Use rubber cleaning gloves to hold the unit so it won't slip. Hold the unit in your left hand and turn the top slowly with consistent pressure counter-clockwise (away from you).
If that doesn't work, you can try taping the case very gently on a wooden surface, while you are twisting the case. This needs to be done while you are physically twisting it. The vibrations can help to dislodge the two sides of the shower filter.