Unsurprisingly, the game was already discussed elsewhere. There is a post on PSE, which links to a longer discussion on The Grey Labyrinth. What you are looking for is in Amy's response to "aqqqqua*" in that discussion.
I've collected bits of information, which I present as increasingly more specific hints.
The riddle ...
... tries to make you think about time and such, but that is a decoy. The real answer is an object (and one that doesn't fit too well with the theme of mythological creatures).
I am forever on the Way, but I will never reach its end.
"On the way" is important here, the rest is misleading. Perhaps it should rather be: "I am (something that you might see) on the way".
I see the Way in three aspects.
The three "eyes" are real objects, and not, for example, instances of the letter I.
With my first eye, I see it as endless motion;
One eye means movement.
with my other eye, I see it as eternal stasis.
One eye means no movement.
Yet I have a third eye, which sees both without contradiction;
One eye means an intermediate state.
The three eyes ...
... have different colours.
What am I?
I think you can guess now who I am.