Any ideas in using outdoor paint to paint the wall around bathtub/show

Yes! Using outdoor paints in your kitchen/bathroom is an old trick I learned from one of my favorite painters. Outdoor paints have all the same mold and fungus protection that kitchen and bath paints have, plus they tend to be even more resistant to moisture.

Only word of warning I have is that outdoor paints tend to be a bit thicker and can be a little trickier to work with.

I wouldn’t use an outside paint as I think a marine paint would do much better.

Just because you "can" doesn't mean you should. I wouldn't. There is a voc concern, not washable like interior paints are formulated to be. A host of no reasons why. Anyone with respiratory problems will be ill. Not good for pregnant women and children or your pets. Nope nope nope.

https://www.bobvila.com/posts/86771-help-exterior-paint-used-inside-my-home?page=1

http://community.homedepot.com/howto/DiscussionDetail/Exterior-Latex-Paint-Indoors-9065000000004wf

http://primetimepaint.ca/2013/whats-the-difference-between-interior-and-exterior-paints/

One this to consider when using exterior paint is to look out for is the VOC’s (Volatile Organic Compounds), or what is commonly referred to as, paint fumes. These fumes will permeate in the room and can last for as long as six months or more if the exterior paint is allowed to stay on the surface for that length of time.

I'd suggest buying a good moisture proof paint for the bathroom. I painted the ceiling of my bathroom with it and when we had ice build-up on the north side of our house and got leaks the full length of the house, the only ceiling that wasn't damaged was in the bathroom. That paint stopped the moisture from coming through from up above.