Sounds like you want a door bell that has two ringer locations so that you can hear it when in the basement, as well as hear it on the first floor.
Several have described a single ringer that is capable of distinctive ring based on button location, e.g. front door or back door. AS I understand it, that is not what you are looking for.
If you are in the wiring stages now, just run another bell wire pair, or most use cat 5 now just because it is cheap and available and you can double up pairs. Run this cable from your bell transformer/ringer location to the place you want the second ringer. The ringers will be wired in parallel and both will sound with a button push.
In my room I have a wired door bell, but a wireless extension ringer in the other end of the house. The wireless transmitter is located where the original ringer and transformer are, and it sends a signal to the remote ringer, which plugs into any AC outlet.