Is random access allowed in the Bit Complexity model, or is it just expensive? In the RAM model, you're allowed to do unbounded indirect access (pointers can be arbitrarily large and still fit in a single machine word).

In the Bit Complexity model (no wiki article, sorry), machine words are constant sized. So storing the address of an entry in a table of size $N$ would require $O(lg(N))$ words.

This raises the question: Are you even allowed to do a multi-word random access into a table of size $N$ when using the bit complexity model? It's a bit of a strange operation, because the machine can't "hold the jump in its head" while making it. On the other hand, it could be allowed but cost $O(lg(n))$ time instead of the RAM model's $O(1)$ time.

I realize that this is basically just a definition, with interesting things on either side, but I'm wondering what the common definition is. Is bit complexity just another way of saying "multi-tape turing machine"?
According to Fürer's paper, the underlying complexity model is a Turing machine without random access, or equivalently, Boolean circuits. So no random access is allowed. Check also Diem's writeup on the subject.