Is mes (the first two letters pronounced the same way as me, an alternate spelling is probably me's) an accepted plural form of me?
There are other mes in other possible worlds.
Yes, it is possible to find it used this way in published works. From Google Books (bolding not in the original):
Each decision that created a subtly different universe, created another of us, another of a nearly infinite number of mes, who added just a fraction more to our intellect and understanding.
– Paul Melko, "Ten Sigmas"
"Me" is being used here as a noun; pluralizing the pronoun "me" gives us "us" instead.
Using "me" as a noun like this is relatively rare compared to its use as a pronoun. But there are still numerous examples of this usage, as can be seen from this Google Books search of the phrase "the me".
The plural, "mes", is probably even rarer, since it is generally not needed outside of science fiction or speculative contexts.
As some have mentioned in the comments, it also has the downside of looking somewhat like it should be pronounced in the same way as "mess."
I think that a more immediately recognizable spelling/punctuation of plural me would be me's. Here are some examples from Google Books search results. From John R. Mayer, Gilgamesh: A Reader (1997):
When I address you, I also address myself. As I observe you I also observe myself. In this observational act, the inside of each of us, our subjective I's, become the outside of us, our objective Me's. As I am a Me to you, I become a Me to myself. When I as a subject speak to you, I also speak to me as an object.
From Anselm Strauss, Mirrors and Masks: The Search for Identity (1997):
The I, as subject, in reviewing its Me's as objects, continually moves into a partially uncharted future; thus new I's and Me's—that is, appraising act and appraised acts—necessarily emerge.
Some of these transactions consist of each person's responses to himself. It would be well here to recollect Reizler's remark that the "I" can respond to many "me's": among them the me of yesterday, tomorrow, several minutes ago, the me of the immediate present, and the me in general. In face-to-face situations, persons respond to various facets of themselves and their performances.
From Christa Metzger, Unemployed: A Journey Through the Dark Woods (2011):
If I had only been able to — alloweed myself to — enjoy such carefree moments more in those days! But my mind was mostly judging and criticizing me for pursuing all these stupid little things to fill the big void left by not having a job. I began to wonder if there were two me's — the one criticizing and the one doing these things. It was as though I was observing myself acting in this different role.
And from Andre Depuis, Of Heterogeneous Disposition (2013):
I could sell my book modestly, if at all, and it didn't matter.
I'm sure there are other 'me's'. Like me, but different.
Better than me at being me.
As the first two examples suggest, one advantage of the spelling me's is that it makes a more natural pairing with I's, which is a much more appealing plural form of I than the alternative Is is.
In any case, I get the impression from Google Books search results that me's is a more common plural form of me than mes is (although both spellings produce numerous false matches for such things as contracted "me is" (me's) and French my (mes). That's the spelling I would use.