The bass clarinet does sound an octave lower than the clarinet, but it uses the same fingering and thus offers pretty much the same basic range. The bass clarinet usually has additional keys to get even more notes on the low end. Generally a woodwind instrument has a standard range and an extended range, which can go as far as technique allows (which we’d call altissimo range). The standard range is quite easy to play, but altissimo range will start to require special fingerings and a good setup and technique. This also means that not all changes between altissimo fingerings are easy to do. But generally with woodwind instruments the lower pitched instruments are nice to overblow and and offer a wide range. This means that although the altissimo on bass clarinet has been ignored for a long time it does offer a very nice altissimo range.
For clarinet we have different registers:
- The chalumeau register from written E3 to Bb4 (on BC the lower note goes lower)
- The clarin register from B4 to C6
- The altissimo register from C#6 to (?), with fingering charts existing up to A7 (for bass clarinet even up to C8)
But these altissimo notes can be hard to play, can be limited in their tone or dynamics, they can be problematic wrt. intonation. Thus you should use them only if you know exactly why you are using them
Also keep in mind that these instruments sound very different in these different ranges. A part played by a soprano in chalumeau can also be played by a bass in clarin, but will have a totally different color.