I'm assuming that you want to make it more difficult for learners to cheat at course work and/or exams. Is that correct?
I think web analytics software could help to identify if the same computer has been used to access several different accounts. Each computer and its software are configured slightly differently which is a bit like a digital fingerprint (not totally reliable though) and you can also store cookies, local shared objects (LSOs = Flash cookies) and/or the latest tracking technique is canvas fingerprinting. Google, addThis, Facebook, etc. all use these techniques to uniquely identify millions of users even when they're not logged in to those services (I remember reading once that Google had over 50 different markers/variables that they could use to identify people with).
There may even be ready-made integrations with Moodle so you can merge and view user data from both.
Tim's right that the most reliable way is by proctoring.
I hope this helps!