I think the question was whether it is possible for a number to be normal in one base but not in another. Intuitively I would say that randomness should transcend the base and therefore a number that is normal in one base should be normal in any other base and therefore absolutely normal. But if that were the case the distinction between normal in a specific base and absolutely normal would be unnecessary. So I don't know but would also be interested in an example in case one exists and is known.

EDIT: It is easily possible for simply normal numbers, i.e. if you only consider single digit frequencies but not frequencies of digit pairs, triples and so on. 0.(0123456789) is simply normal in base 10 because every digit occurs with frequency one tenth but it is not normal in base 10 because only ten out of one hundred digit pairs occur. But it is also not simply normal in base 10¹⁰ because it then consists only of the single digit representing 0123456789 repeated indefinitely.