Turing machine with a countably infinite alphabet Suppose you have a Turing machine, but with a countably infinite alphabet.

How does such a Turing machine differ from an ordinary Turing machine, either in terms of computability or of complexity classes?

Do complexity classes for ordinary Turing machines hold up in such a model?

Edit: quick addendum - it seems obvious that the existence of a countably infinite alphabet leads to uncountably many state transition functions. I'm curious for the answer to both my original question, and to the restriction in which the only transitions allowed are recursive.
Your model accepts all possible languages over the countably infinite alphabet.

The first step is to convert the input word to a single symbol, in some one-to-one fashion. We can do that since $\bigcup_{n=0}^\infty \aleph_0^n = \aleph_0$. You can then encode your language in the transition function.

More generally, to get a non-trivial model you need to ensure that the description length of the machine is smaller in cardinality than the number of words over the alphabet.