I think that if you use by or next to then they would both give the impression that you are sitting close to your computer. However, next to is more specific.
The preposition next to means to the side of and the use of by can mean near to something or somebody.
E.g. I sit next to my friend in the classroom.
= the speaker is beside her best friend,
E.g. I am sitting next to my computer.
= I am sitting to the side of my computer.
.E.g. His desk is by mine.
= His desk is close to the speaker's desk and not necessarily to the side of it.
E.g. I am sitting by my computer.
= I am close to my computer, but not necessarily to the side of it.
People also say at their computer which is usually taken to mean you are sitting or standing facing your computer and probably working at it as your picture shows.