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.

Neosporin is an antibiotic, used to prevent or treat minor infections. If the bruised area included broken skin (a cut, scrape, puncture, or burn that compromised the barrier the skin provides) then applying neosporin or another topical antibiotic could be beneficial

There is no reason to apply an antibiotic to a plain bruise

National Domestic Violence Hotline website as this line of inquiry is potentially concerning

If your body is otherwise healthy then you should just let it do its thing. If you're not otherwise healthy then the list of possible answers varies (into opposites) depending on what else is wrong. Example: (as I understand it) a warm bath should dilate blood vessels in the skin. A cold bath will contract surface vessels and dilate deeper ones. Hot and cold aren't the only things that will dilate/contract blood vessels though -- a bruise for instance...

A bruise is a result of damaged cells. There will be cleanup and rebuilding going on. Conscious Interference is unlikely to help unless you know exactly what to do, but in that case you probably also knew what to do to build a strong healing response so just relaxing would be your best bet.

Anything that increases blood flow (hot baths, exercise, sunlight,etc) will increase the inflammation response. This is why fevers are a thing and generally why being static and immobile causes wounds to take forever to heal

I've read studies about using leeches to help "clean up" bruises more quickly. Of course you then have the bite mark to contend with...