Mother-in-law is 94 with serious dementia, and have agreed with her daughter to have mom for alternating months. Will this cause more confusion?

Will changing houses cause more confusion for MIL? Yes.
Is switching back and forth every other month a fair distribution of work? Yes.

So many things are 'redone or rerouted' for the convenience, comfort and ease of the dementia victim, that really puts a strain on the caregivers. Many caregivers (me included) are hanging on by a thread, (physically, emotionally, financially, spiritually). Unfortunately, sometimes, to keep the c/g's in good health, the choice must be made in favor OF the c/g. This sounds like one of those cases.

You stated your loved one is very confused anyway, so a division of the labor, so as not to exhaust only one person, would be the best of both worlds. Your loved one will adapt and you will be refreshed after a 1 month vacation (every other month).

I vote to go ahead and give it a try. There is no sense killing the one caregiver to keep the continuity for the patient. In the end, the c/g won't be giving good care when they are exhausted and broken.

Just my 2 pesos.

Hbcoble, you are correct, switching Mom-in-law from one home to another every month will add to the confusion.

Best to leave her at one house, but it wouldn't be fair to the person who has her full-time, and it's not easy to co-help at one home as we all do things differently and that could cause a major rift within the family. Can the family get together and use Mom's social security or any other retirement fun to hire a caregiver for a few hours per day?

Can Mom-in-law budget to move to Assisted Living/Memory Care? The cost isn't cheap, around $6k per month depending on your area. That way Mom-in-law would be around people of her own generation, and have 3 shifts of caregivers.

But if her dementia is serious as you had mentioned, and moving to Memory Care isn't in the cards, have her apply for Medicaid [different from Medicare] where she would go into a Nursing Home and Medicaid would pay for all of her care. I know that isn't always an option for some families as they want Mom at home. Just a suggestion.