Just because it’s not possible on a Turing Machine doesn’t mean it’s impossible on a PC with finite memory. You just have to track all the memory that is available to the algorithm and once you detect a state you’ve seen already, you know it’s not halting ever. The detection algorithm will need an insane amount of memory though.
Edit: think about the amount of memory that would need. It’s crazy but theoretically possible. In real world use cases only if the algorithm you’re watching has access to a tiny amount of memory.