In Avengers: Age of Ultron (2015), shortly after Vision manifests, he looks at Thor, notices Thor's cape, and generates one of his own.
Why did Vision care to copy this aspect of Thor's appearance?
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In the scene where Captain America and Maximoffs trying to stop Stark and Banner from creating an another Ultron, Thor appears and shot a thunder blast into the container that Vision was held in. By doing that, ultimately Thor helped waking up Vision, and after waking up, the first person he sees is Thor. Combining the two facts, Thor was the most recognizable person for Vision in that moment. Then he jumped onto Thor and was thrown away. In front of that glass window, he stopped for a moment, gathered his thoughts and made himself wear some kind of greenish suit. Then coming back, he sees Thor is wearing a cape. So Vision did what every kid would do. He copied Thor because he was the most recognizable and closest as a person in world for him.
As others have suggested, it could be also the reason that Vision in comics usually wears a cape.
When Vision comes to life he attacked Thor first, so maybe this was his way of showing gratitude towards him as Thor was responsible in his creation. Also remember cape is a iconic part of Vison in comics, so maybe they have to do it in a reasonable way possible.
It was fully opposite in deleted scene where he did it to compete with him:
Vision had a mind stone on him. It was one of the six infinity stones and had great power. The power was that Vision could read the minds. Just when he was born, he was not fully conscious and seeing a gigantic person (Thor) beside him in an attacking mode, he attacked. But then, when he stopped after being thrown by Thor, he waited and looked at himself and realised. Not only about himself but about everyone else also as he could read the minds.
So, when he returned he said -"This was odd" because having a mind stone and attacking even though you know that everybody has good intentions is odd.
Then having read the minds, he realised that Thor had a role in bringing him to life, and he said him thank you. He also realised that he is closest to Thor (as both are not human) and only Thor has the flying capacity (natural, not like Iron Man) so seeing all these similarities and in honour, he manifested the cape. I hope this answer satisfies you.