Neosporin wouldn't help. A bath may increase good blood flow but probably won't improve on the bruise healing to a noticeable degree. Ibuprofen can make blood thinner and increase bleeding so would be counterproductive at least right away. I don't know what if any benefit steroids would have.

Bruising often accompanies injury or trauma - some treatments may decrease pain and swelling making the area look better but not necessarily healing the bruising faster.