When trying to setup my new laptop, I keep getting the message:

That Microsoft Account does not exist. Enter a different account or get a new one

I use Office 365 E3

If I say get a new one and enter my Office 365 address, it says:

You can't sign up here with a work or school email address. Use a personal email address, such as Gmail or Yahoo!, or get a new email address

I want to use this laptop with my office account - why can't I?

Is this the wrong version of Windows or something?

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Yes it was the wrong version - Windows 10 home - I had to upgrade to get it to work, which was expensive-.

Now I know to be careful when choosing my windows version. ‘Home’ means ‘not for any work use at all!’

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Yes the best option is to use an outlook.com account and then enter the new serial or upgrade online. After that you can rest the PC under control panel > recovery and then rig up the system for your Office 365 username as if it were new.