We always get work done, regardless of what mood Bill or I is/am/are in.
Which form of 'be' would be used here?
In this case, two subjects are connected by 'or', you use the verb form of the subject that is nearest the verb - 'am'. We always get work done, regardless of what mood Bill or I am in. Google 'Subject verb agreement', you can get more detail.
Let me just quote directly from Huddleston & Pullum's Cambridge Grammar... (p. 509)
Where one of the coordinates is 1st or 2nd person singular there tends to be a preference for agreement with the final coordinate, except that the 1st person singular form am is felt to be awkward and the construction is likely to be avoided in monitored style:
You must'nt go unless [either I or your father] comes/?come with you.
I dont think [either your father or I] have/?has had much say in the matter.
[Either your father or I] ?am/?is going to have to come with you.
So despite my critical comment on one of the other answers, they are saying that in general proximity agreement is usually the better sounding option when the subject is or-coordinated. Except not when one of the coordinates is I.
What is interesting is that in the third example they gloss both alternatives with the question mark (meaning it sounds awkward). So the official answer from the currently most authoritative book on the subject is... there is no good choice. For what it's worth, is sounds a little better to me.