How can I install a grounding rod with bedrock four feet down?
You can drive at an angle. If I remember correctly, you can dig a trench and lay the 8 foot rod(s) flat in the bottom of it (Personally, I'd never go with one rod anyway.) If you wanted a really good ground, you simply excavate your foundation to bedrock and use an UFER ground in the foundation concrete (it's a concrete encased ground electrode formed by electrically connecting the reinforcing steel in your concrete foundation - or a section of copper cable, but the steel is there and serves multiple purposes.) 20 foot minimum, but common sense is to tie all the steel together for the best possible result.

Note on re-reading - A CCE/Ufer ground does not require excavating to bedrock, but if bedrock is 4 feet down, you might was well put your foundation on the bedrock...

Responding to erroneous comment below:


  Exception 1 to 250.68 indicates that buried electrode connections and
  concrete-­encased connections to electrodes shall not be required to
  be accessible. Exception 2 indicates that exothermic welded
  connections and irreversible connections to grounding electrodes that
  are encapsulated in fireproofing material are not required to be
  accessible. The exception applies not only to the connection but also
  the mechanical fastening means, such as a nut or bolt, that
  establishes the connection to the metal framing member.


From EC Mag article. But you can also look it up in the code, that was just faster to find.