Is there a native notation for base 60/sexagesimal numbers? I have been reading about the Babylonian-derived sexagesimal or base-60 number system. Is there a standard notation for base-60 numbers that is analogous to that used by binary (0-1), octal (0-7) and hexadecimal (0-9,A-F)?
It looks like many people use a colon-separated notation that is similar to digital clock readouts (e.g. 33:20:06), but in that case, decimal numbers are being used to encode sexagesimal digits rather than those digits being expressed in a native format. This practice is also referenced here on Math.SE. Even the  sexagesimal digits (sexagesits?) as they were actually used by the classical Babylonians appear to be composed of underlying decimal digits rather than a purely base-60 system of 60 unique sexagesits.
Does a "native" system of sixty different number symbols for sexagesimal numbers exist, whether or not it has ever received widespread usage?
To be clear, I know I could create my own idiosyncratic base-60 notation system using a combination of decimal digits (0-9), the letters of the Roman alphabet, Greek letters, Hebrew letters, smiley faces, frowny faces, squiggly lines, emojis, cartoon characters, caricatures of (in)famous politicians, etc., but my real question is if any serious such system actually exists in mathematical literature, pedagogical resources, or historical epigraphy.
I'll be monitoring this question for the next ג☺️4fΩ minutes.

Mesopotamians (actually, Sumerians first, then Akkadians of which Assyrians and Babylonians are dialects) used a simple system of bars, stacked if more than 3, with a different mark for 10. So that would be
 = 1,  = 2,  = 3,  = 4,  = 5,  = 6,  = 7,  = 8,  = 9 (although there were variants of most of them; keep in mind cuneiform was used for about three millenia!)
Then we jump to  = 10,  = 11, …,  = 20, …, but  = 60. Yes, that could be confusing, with only context helping to understand if  meant 1 or 36, or 3,600, or 216,000 or… (you get it).
As far as I know, there is no modern system with different characters up to 60.
By the way, to indicate fractions, it is now customary to use a semicolon, and most sources I’ve seen (not to brag, but quite a few!) use commas to separate groups, so for example one would have 11,18,32;29,5,40 for (11 × 3,600) + (18 × 60) + 32 + (29 / 60) + (5 / 3,600) + (40 / 216,000) = 40,712.4849074074074074…
