For example, using Chaitin's constant as a base. I assume this wouldn't be very useful for representing any form of computable number, as any computable number would most likely become uncomputable and completely random. But can you use an uncomputable but definable base to actually do anything interesting? For example, does this mean that computability is relative to something and not absolute? Because Chaitin's constant would be trivially computable in a base Chaitin's constant number system, whilst 5 would be uncomputable I think, I cannot prove that however.
Edit:Chaitin's constant+1, Base n numbers where n is between -1 and 1 are even weirder and integers seem to become undefinable. And also to be specific(the "constant" isn't just 1 number) Lets just say, for example, in this case, The probability that a completely random,finite, Turing machine will halt on a completely random, finite input, but you could use any definable, uncomputable number for this.