It seems to be fairly easy to construct normal numbers to a fixed base $b$. For example, append all integers in base $b$ to $0.$, or all primes, or all squares. On the other hand, it seems very difficult to construct numbers that are normal to all bases $b$. In fact, it seems that Chaitin's constants are among the more accessible examples.
Is it (relatively) easy to construct numbers that are normal to two (coprime) bases? If not, what are the significant differences between trying to make it work for one base versus two bases at the same time?