Can you have a natural swimming pool indoors?

Yes, it is possible to build a natural swimming pool indoors.

The principle of natural water filtration is the same as for outdoor swimming ponds i.e. to keep the nutrient level low and in balance.  There are some additional design factors to take into account, for example using appropriate materials for high humidity, having enough light for the plants (if plants are being used), and having the right temperature for the filtration to work effectively.  All these factors can be covered at the design stage.