Generally, it is incorrect to think of a primer as something to use when you are trying to help cover up another color. This is a very common misconception, kind of like an old wives tale. This tale will continue to live on though because the clowns that work at Home Depot keep telling us to use a primer when you are trying to cover up a dark color. Primers do not cover well. Their job is to seal, to stick and to provide a good base for your topcoat. In your case, it sounds like the maroon color is already dark so covering that up with another dark color should be a cinch. If you were to use a primer though (for some other reason) then it would be a good idea to have your primer tinted towards the finish color because it would be going backwards to prime with white. The purple will cover the red much easier than trying to cover the white.

So I wouldn't think I need a primer for another reason? The paint that's on the wall is eggshell finish and we would just be adding eggshell finish paint to what's there---the depth of the color is pretty similar I think. So it would be as if just adding another coat of paint I think?

you are keeping withing the same tint bases it sounds like, so you should be fine to go ahead with your new color and also you mentioned your previous color was an eggshell sheen.

I would however suggest making sure your walls are clean and dust clean; after all this is is a bathroom that gets frequent use and is subject to sprays, lotions, oils, toothpaste, zit cream on the wall or cleaning materials that may have bounced over on the wall where the toilet is . Believe it or not I have been in homes to redo the baths and well lets just say if there are little ones or may have missed the toilet . No offense men, but it does happen. :) I have seen quite a few head scratchers in my time.

I am sure you have already taken these precautions and are aware of the pre prep.
good luck.

It is not necessary to prime each time on a job. Every job needs to be evaluated on its own findings to determine a primer and the type of primer to use. :)

That's exactly right rosewest!

Here is the thing though. Some people claim that using a primer for every single paintjob is the best way to go. It may well be...I certainly can't say that using a primer every time is a bad decision. But, most of the people claiming this are trying to sell primer! Plus, let's say that I have compared paintjobs...a painted wall that was primed first seems to be as durable and as long lasting as a painted wall that was not primed. Once you have arrived at maximum saturation (enough coats so that no more paint is actually soaking into the surface), using a primer really seems to be a waste of time to me. The only exception to this may if you were painting with red over white....then you may tint the primer grey. The primer is cheaper than the topcoat most times. But, since Aura has arrived and reds cover anything with that product in two coats, we don't even mess around with the grey primers anymore.