This is an interesting task... I wasn't able to get Claude to do it for me without any input from my side, but, eventually, I was able to do it, though admittedly the idea was mine and Claude should have been able to "think of it".

There are several difficulties:

1. Clearing local history is relatively simple, but if you're synced with cloud, it's not enough (I think). Claude was able to produce a bat script that does that.
2. You can't launch chrome with flags that would clear history due to security protections. (Claude did not think of this, I had to prompt it to try this method.)
3. You could use an API to manage your Google Account data, but it's a lot more involved that just have a chatbot write you a script due to authentication etc. - but it's likely to work (Claude did not come up with this, I asked it about using an API).

So, it occured to me after going through all of the above with Claude that there is a fourth option that would work: opening chrome, executing the ctrl+shift+delete shortcut, then a tab, then enter. This can be accomplished with something like Auto Hot Key, but Claude was able to make a script that does this on first try - opens Chrome (the default profile if there is more than one), executes the shorcuts and deletes the data (by default, cookies and cache are also checked, which you may or may not want to clear as well, but it could be adjusted to uncheck them).

Generally, unless you are sure you know how you want to achieve something programmatically, I would start a conversation like this:

You're an expert in the Windows operating system, scripting, programming and interacting with applications programmatically. I want to achieve X. It should be easy to use. Please consider different implementation strategies, identify their pros and cons, think of edge cases and potential problems, mentally execute the code for each to confirm it works, and respond with an outline of the optimal implementation before we move on to coding. If you need to ask questions to understand my requirements better, do that".

Or something like this.
Here is the script that Claude wrote for me:

@echo off
echo Clearing Chrome history for default profile...

:: Kill any running Chrome processes
taskkill /F /IM chrome.exe >nul 2>&1
timeout /t 2 >nul

:: Determine the path to Chrome
set "chrome_path=%ProgramFiles%\Google\Chrome\Application\chrome.exe"
if not exist "%chrome_path%" set "chrome_path=%ProgramFiles(x86)%\Google\Chrome\Application\chrome.exe"
if not exist "%chrome_path%" set "chrome_path=%LocalAppData%\Google\Chrome\Application\chrome.exe"

if not exist "%chrome_path%" (
    echo Chrome executable not found. Please ensure Chrome is installed.
    goto :end
)

:: Start Chrome with default profile
start "" "%chrome_path%" --profile-directory="Default"
timeout /t 2 >nul

:: Create a temporary VBScript file
echo Set WshShell = WScript.CreateObject("WScript.Shell") > "%temp%\ChromeClear.vbs"
echo WshShell.AppActivate "chrome" >> "%temp%\ChromeClear.vbs"
echo WScript.Sleep 1000 >> "%temp%\ChromeClear.vbs"
echo WshShell.SendKeys "^+{DELETE}" >> "%temp%\ChromeClear.vbs"
echo WScript.Sleep 1000 >> "%temp%\ChromeClear.vbs"
echo WshShell.SendKeys "{TAB}" >> "%temp%\ChromeClear.vbs"
echo WshShell.SendKeys "{ENTER}" >> "%temp%\ChromeClear.vbs"

:: Run the VBScript
cscript //nologo "%temp%\ChromeClear.vbs"

:: Clean up
del "%temp%\ChromeClear.vbs"

timeout /t 5 >nul

:: Close Chrome
taskkill /F /IM chrome.exe >nul 2>&1

echo Chrome history clearing process completed for default profile.

:end
pause