Is there a way to contact specific user?
Generally speaking, no. Stack Exchange is not a social network, and does not aim to be. Hence, unless a user discloses contact information, there is no easy way to contact a specific user either inside or outside of the SE network.
Is there some means of continuing such discussion somewhere else than in comments or chat?
I am afraid of using chat because I know that I am from Europe, while the user may be from USA, which means about 7-8 hours time difference by default.
Chat is asynchronous. You can leave a message in chat, and the other user comes by later, reads it and responds. In that regard, chat works in exactly the same way as comments.
Assuming that you feel the conversation is useful, but not necessarily where it is currently located, I would recommend just creating a chat room and moving the conversation there. Just because it's chat doesn't mean it has to involve everyone being present at the same time. Create a chat room, copy all previous comments into it, and leave a link at the end to it like "let's continue this over here: [link to chat room]". Flag the post for moderator attention and request a comments cleanup. All done.