I would compute and interpret the importance scores within the context of each branch of the alternative-specific design.
For example, if my design involved bus, car, and train. Each alternative has its specific attributes that only show up with it. Also, there may be a few generic attributes that apply to all three alternatives.
I would take the raw HB utilities into your favorite analysis program...it could even be a spreadsheet program like Excel. Then, compute the difference in utility for each attribute for each respondent (except for the principal attribute which contains the levels bus, car, and train). Next, summarize the importance scores for Bus, by making the range of utilities for each attribute that apply to bus (including any generic attributes) sum to 100%. Do the same for car and train.
This makes the importance scores sum to 100% for the bus decision, the car decision, and the train decision. You interpret the importance scores only within each of the principal alternatives, such as "assuming respondents are considering choosing the bus, what is the importance breakdown within the attributes that apply to bus that drives their decision"?