Cronbach's alpha is not relevant to CBC. It is a measure of the internal consistency of a set of items purportedly measuring the same underlying construct. CBC questions are choices from which utility measures are derived, they are not themselves measures of any entity.
Sometimes folks repeat specific CBC questions twice in a given respondent's survey, and in this case you could measure a type of reliability called test-retest reliability, but of course that is a different measure than alpha.
Another measure folks use sometimes as a measure of reliability is the scale factor of the logit model. This is pretty easy to do for groups of respondents, but across individual respondents in a single study it is impossible to disentangle reliability/consistency from the random error component of utility.