Yes, the CompleteFTP service can run under an AD user account. In fact, this is advisable in scenarios where non-Windows users need to access network shares on other machines, since it allows permissions to be controlled more neatly.
You need to ensure that the AD user has full control over C:\ProgramData\Enterprise Distributed Technologies\Complete FTP , its subdirectories and any other directories that the CompleteFTP virtual file-system maps to. The AD user doesn't require permissions for directories that are only accessed by Windows users. The AD user should also have permission to log on locally, otherwise Windows users may have problems logging in through CompleteFTP.
Obviously you need to test it thoroughly before putting it into production.