There's "!?" and "?!"... I think those are both are more descriptive than one character.
My new annotation symbol: ^ for "surprising move".

I should have mentioned that because ^ and the conventional annotation symbols are independent, that means they can occur at the same time. Therefore unexpected combinations that are sound could be marked "!^". Tal's combinations, which weren't always sound, could be marked "!?^", since they warrant further analysis and probably shocked Tal's opponents out of their socks, besides.

Seems like it's value would be limited to one's personal notes, since what is surprising to one person may not be surprising to others. I will give it a try in my own notes and see if it helps me remember.
True, but then the standard annotation symbols are also subject to opinion, unless you're a computer calculating centipawn differences. I'm glad to hear the notation might be useful to you.

I agree with my friend Mick, !!!!! and ?????? are more descriptive with less characters, less is more/better in language...and further with the use of !!!! surprise is already implied.

I played an old guy in Indiana about 20 years ago who put all sorts of odd symbols after every move. He also played the Bird variation against the Ruy. I could never tell which led more to his downfall.

I have mixed feelings about this. If an unexpected move is the best, then ! should suffice. The exclamation mark is usually used for an unexpected good move, not just a good move (which is why 1.e4 e5 2.Nf3! is not standard annotation, even though we all know 2.Nf3 is the best move in that position). If an unexpected move is not best but still decent, !? seems to cover that completely. If it is unexpected AND not as good as regular, normal moves, ?! fits the bill. It doesn't seem necessary...
... and yet it does in a way. I'm thinking of opening lines where two moves may lead to the same position, but one move gives the opponent more options. These options may be sound or unsound, doesn't matter, that's more to prepare for. Using ^ or some other symbol to show, practically, that this is an easier choice could be useful.
It could also be useful for self-analysis. You could go through your games and annotate whenever your opponent played a (good!) move you didn't expect. After a few weeks, you would have a large collection of positions to look at, positions where you miss something, and you can use these to train yourself more effectively. As a personal annotation, meant for your own training, it might not be bad.

If an unexpected move is the best, then ! should suffice.
What you and three earlier responders are saying is that one symbol like "??" or "!!" already implies "^". No problem. That does show that the concepts aren't completely independent, but as you said, there's no need to add every possibly applicable symbol when there exists an obvious dependency. For example, even though we can always write Ngf3 instead of Nf3, or KP-K4 instead of P-K4, we don't do that because the added information is redundant. We're already following such non-redundancy conventions, even if we don't think about it, so that complaint of redundancy is not a sound argument.
Annotations are optional symbols that are there to help us if we want to use them; annotation symbols don't come with rules that say you must use them. If you don't find annotation symbols useful, then don't use them, whether they're standard, mine, or your own.
Recently while writing down some lines I was exploring and trying to remember I hit upon an interesting idea as a memory aid: use a new annotation symbol for a surprising/unexpected move. The practical problem I having was that some book lines were surprising and counterintuitive, so they would have been hard to remember or figure out over-the-board, and at the same time they weren't the best lines, either, so the common annotation symbol "!" for "best move" wasn't accurate, either. Neither was the symbol "!?" for "interesting move" since the book move I was considering had already been well evaluated so it did not qualify as a line in which to invest further investigation later, which I assume is the intent of the "!?" symbol. In other words, my new insight was that "surprising" was a different concept than "best" or "interesting" so it should have an independent symbol as well.
I studied my keyboard for another symbol to use, and about the only sensible symbol not yet used that couldn't be easily misinterpreted was "^". This wasn't too bad, especially since the emoticon using that symbol meant "joy" (https://en.wikipedia.org/wiki/List_of_emoticons), which wasn't too far off in expression.
I liked this idea enough that I began implementing it immediately in my repertoire, and I'm still using it that way. This is sort of a nonsense post, but at the same time it could be a useful enough idea for enough people out there that I thought I'd share it. Below are some examples of where I thought this symbol would be useful. It is often used when there exists a zwischenzug in an opening, for example, since a zwischenzug is often quite an unexpected move, which has that surprise attribute. That means the symbol might also be used to search your repertoire text for zwischenzugs if you don't already keep track of those.