Bedrock in Basements
There is a reason that foundation are built on top of bedrock instead of in bedrock: The thick layer of rock cannot be moved as takes an incredible amount of work to cut and or change at all. Bedrock layers are left as they are to provide a high level of stability to the foundation constructed over them. Every house is different, but you may find that a quick survey will show cutting bedrock is not possible or practical, even if it can be accessed through your basement level.
When your basement is first being constructed and is built directly on the bedrock, some bedrock work is common. In this case, you can cut into bedrock and remove sections to make way for your new foundation, leveling out the bedrock surface to accommodate your home. This is the easiest way to remove any of the bedrock because it is still exposed. This also provides a chance to drill down and install rebar in the bedrock for extra foundational support.
If your basement and home are already built, excavation into bedrock can be much more complicated. You should contact a surveyor or construction company and ask them to take a look to see if your bedrock can be removed to enlarge or change your basement area. Slight changes may be possible, especially for extra foundation support like added rebar. But excavating in a pre-built basement can be very expensive, even without dealing with bedrock. Even if the project is possible, it may not fit within your budget.
Wells are sometimes constructed on top of bedrock. The bedrock layer is "cracked" slightly to allow the well to re-fill with water more easily. If you want to install a well in your basement but you are not sure how easy it will be to cut into the bedrock, surveyors and specialists can offer invaluable advice here, too. A bedrock well may be possible, but anything from the position of the foundation to the use of fertilizers in your garden may make drilling a well in your bedrock inadvisable.