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It's actually one of the oldest traditions in storytelling. Many Greek plays and stories left without firm resolution.
My favorite example of this style is The Graduate. At the end, Dustin Hoffman has escaped with the bride and sit in the back of the bus. They camera lingers on the two as they smile at each other and then the smile fades as they stare back into the camera.
Do they stay together? Do they split? The movie doesn't say. It asks the viewer to make up his/her own mind. The movie provided resolution on the actions of the characters, but does not completely resolve their arc.
No Country for Old Men is similar in this regard. What "end" is there in the cycle of drugs and violence? And as far as Anton Sigur is concerned, I suppose there could have been more resolution, but since the story focused on Tommy Lee Jones' sheriff, I don't think it's called for. Jones' sheriff is past his time and his capabilities to catch Sigur. So it makes sense that the story ends how it does.
In summary, I think the "no ending" as you put it is better if the story actually warrants it. However, leaving off a firm ending that wraps everything up shouldn't be a crutch to make your film feel more "artsy".
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