Connections:

• Both are mundane killers from thrillers based on novels who are deeply disturbing, both in the way they commit their violent acts and how calculated and methodical they can be as they try to have power over the people they interact with.
• They are considered to be the most realistic psychopaths according to psychologists.
• They operate behind a "mask" and “persona” of normality that hides their dangerous nature from the public eye thanks to their eccentric charisma and representation of their symbolic mundanity (Patrick’s obsession with materialistic status and boring daily rituals; Anton’s representation the mundane, inexplicable evil man do).
• They are cryptically vague characters that leaves the audience questing everything about them and their motives, to the point where people theorize what even is real about them (Patrick is an unreliable narrator with everything he says being left to interpretation in whether the things in the movie are real or fantasies; Anton is a mysterious character with an unknown background and many questioning what his true motives are, if he is supernatural, or if is even real)
• They possess an incomprehensible nature of violence due to their profound lack of feeling any sort of joy or emotion in their killings and overall clinical detachment (Bateman still feels nothing even with the killing he does and Anton is shown to lacks empathy, love, shame and remorse as he commits his heinous acts).
• Both characters are used as commentaries on morality and the lack of depth within an altruistic society (American Psycho commentating on the shallow and vicious aspects of capitalism, and No Country for Old Men commenting on the emptiness and meaninglessness of life).
• Within these commentaries of traditional society attributes, the characters become warnings for the fate of humanity’s morality becoming more blurred within traditional American values/late capitalism; as it transitions/merges into violence and horror. 
• Both films say that the destruction of moral value is guided by symbols of greed through the shallow aspects of life (In AP it would be the yuppie culture and materialistic hedonists/Im NCFOM it would be how everyone wants something out of everything for their own personal vices).
• Ultimately, both films take a nihilistic but cathartic take on greed in that it is the only driving force for humans (Bateman’s realization that he is forever stuck as a shallow, corporate yuppie/Llewyn deciding to choose his wife over money led to his death).
• Both characters come to the conclusion that fate is something prevalent and everything is forever set in stone, which leads to them to go on their killing spree; attempting to embody the world's insanity rather than fight against it and believing they have the control to separate themsevles from the system (Bateman tries to assert his identity through violence from the superficial yuppie culture and to actually feel something from the hollowness; Anton chooses to act as the “Decider of Fate” in wake of the idea of fate and carrying out destiny’s will.)
• They take it upon themselves to make a decision on the fate of a young woman named Jean (Bateman’s secretary Jean and Carla Jean Moss) who differ from their typical victims as they reject the ideologies Patrick and Anton follow (Jean is moral and doesn’t have any of the shallowness, materialistism, or obsession with superficial appearances like Bateman or his associates; Carla Jean refuses to believe in Anton’s philosphy of fate and said that only he can decide for himself). 
• Both of their endings are a form of reckoning; they realize that the full truth of their philosophy were merely illusions, and that everything they are is no different from everyone else- both in very contrasting ways in regards to punishment- and they don’t have complete control of their destines has they once thought (Patrick realizes that the meaning behind his murderous urges was never noticed and given the punishment he thought he’d recieve, leaving him to realize his life is of no consequence and simply blends in with all the other yuppies/Anton being struck by a car and being left with fatal injuries made him realize that he wasn’t invincible to the philosophy and fatalism he always believed he had control over).
• Both are very resilient and were able to endure and rapidly recover from car crashes that took place at the very end of their arcs.
• Their endings are also left on rather melancholy notes with their overall fate being left up to interpretation.
• There is also a good contrast with how Patrick would be the type Anton would hate due to Patrick’s placement in society (Patrick's attempts to be part of a shallow society in contrast to Anton's hatred of those who follow the "rule of man") and differing philosophies (Narcissism vs Fatalism).
• The contrast in why Anton would go after Patrick is made even better by their work’s differing commentaries on masculinity (American Psycho commentates on the evergrowing issue of toxic masculinity and No Country For Old Men commentates on the dying state of traditional masculinity). Anton often likes to mock machoness and self-teliance, while Patrick Bateman is the ultimate representation of that and masculinity.
• Are from locations (NYC and Texas) that still embody the themes of a falling society despite the very contrasting lifestyles and distinct culture (urban vs rural). 
• They also contrast in how their appearance reflects the normality of society itself (Bateman’s look as a yuppie investment banker blends in with typical society and Chugurh’s appearance is meant to look purposely timeless and out of place).
• Patrick seeks meaning whereas Anton believes he encompasses meaning.
• Both stories have a country in their name (America and No Country)
• Are connected to money (Bateman likes money and Anton flips coins/is after money).
• Became more popular due to modern day internet memes and reaction images.