I had occasion as of recent to re-watch both of these movies, which date from just about 10 years ago, give or take one.
Dark City was released in 1998; The 13th Floor in 1999. Both are infinitely superior, in terms of story-telling, characterization and asking existential questions, to the pretentious Matrix (1999) and its crap sequels.
Often, as is demonstrated by comparing the three movies, in order to tell a story and make it have a point, you really have to take it easy on heaping up too much philosophical claptrap, even if it's wrapped up in CGI Kung Fu scenes.
If you're looking for superb impressions of existential angst and despair, you won't find them in the overblown in-your-face sequence of Neo emerging from his goopy enclosure—but you may indeed find something clamp tightly around your chest when you follow Douglas Hall, as he drives out of the city and across deserted country roads to the end of his world, and faces the reality of his own artificiality and the destruction of everything he believed about himself; or Inspector Bumstead's numb and futile groping for his memories of a place called 'Shell Beach', or how to get there; and yet it's a place everybody knows—but can never get to.
Maybe the most desolate line is uttered by 'Douglas Hall' (13th Floor), after he realizes the truth and confronts a 'downloaded' person in his artificial world, who tells him that he does, despite everything, indeed have a 'soul'.
"...how can I?...none of this is real. You pull the plug; I disappear; and nothing I ever say, nothing I ever do, will ever matter."
You can't get it more existential-angst-ish than that.
If you haven't seen either of these flicks, treat yourselves to them. If you can't find them at video stores, just download them from your friendly neighborhood p2p network instead. Both deserve the appellation 'classic' and not having seen them is a definite loss.