The way your question is phrased, the only possible answer is to test, just as Izquierdo says. I'd use other types of tests like usability tests and probably measure with https://en.wikipedia.org/wiki/Hick%27s_law , but the bottom line is you'll need to test it.
If your question refers to the need for clues to search actions (whether they are symbols or not), then the answer is yes.
Look at an example below. I started typing "aut" into my Firefox search bar and it shows me 2 previous searches I have done, and suggests more possible searches that will be refined as I type more characters.
Now, the first 2 with the clock icon (as mentioned by you) give me very important information: This is a search I have already done. And it also saves me time: I do not have to type everything, I just click on the suggestion.
And again, you can go even further. Check this: instead of icons, they add photos and faceted searches inside a search flow
You could also check a lot of examples for https://baymard.com/blog/autocomplete-design , and you'll see some of them have icons, some have other methods, but all of them add some clue to transmit more information to the user.
Also remember: icons are informational but in some cases are decorative. In the case you describe, they wouldn't be accessible. Furthermore, they would be invisible for some users.
In short
Yes, what you say is a good idea. It provides useful information, saves time, and shows the user that the different suggestions have different values (visited, not visited, suggested, auto-completed, etc). However, if it's within your possibilities, I would recommend also adding an option that does not rely only on visuals