In theory Life could exist with a totally different basis - however the number of possibilities is probably relatively small - this is why it makes sense for us to concentrate looking for a water based life because we know that that is possible.
In recent years we have discovered that life forms called extremeophiles
http://en.wikipedia.org/wiki/Extremophiles can exist in the harshest conditions some have even been found in nuclear reactors!
However these are very simple organisms whether something complex could exist in these sorts of conditions is more doubtful.