Transferring to Columbia is a very high reach. I would put your chances in the low single digit percentages. If you take two quarters off at Chicago and apply to transfer to Columbia, the most likely scenario by far is that you will be rejected by Columbia. What would be your plan then? Would you then return to Chicago after having taken two semesters off? Taking a gap can be valuable for some students, but this plan, at least in my opinion, is not likely to result in anything other than a gap plus a return to Chicago (unless you also apply to other schools where admissions is way more likely compared to Columbia).
Letters of recommendation are very important when applying to high level universities. They get a LOT of applications from students who are academically qualified to be there. They have to differentiate somehow from among the long list of qualified students. LOR’s are one way to do this.
I have to wonder why you want to transfer. Both are excellent universities. Both have very strong reputations. Both are in large cities. Both will be full of very strong students. Both will be academically very challenging. It will be very difficult to stand out in class at either school. It might be equally difficult, or might be even harder to stand out in class at Columbia just because it is bigger. Frankly between Chicago and Columbia I would be hard pressed to know which is higher ranked overall or for your major without looking it up online, the rankings could switch between this year and next, but I would expect both to be very highly ranked and very academically demanding for a wide range of majors.
Generally in order to transfer to a highly ranked university you should show some compelling academic reason to transfer. Can you explain to us what this academic reason would be?
I have noticed that both Columbia and Chicago actually have more graduate students than undergraduate students. Completing your bachelor’s degree at Chicago and then applying for a graduate program at Columbia is an entirely reasonable thing to do. Most of the people who I know who have a graduate degree, or who are currently studying for a graduate degree, either got or will be getting their last degree at a different university compared to where they got their bachelor’s degree.
But the big question is: Why do you want to do this?
Edit to add: IF you can explain a compelling academic reason to transfer then I might guess that your chances of transferring might get up into the low double digit range, like 10%. This however is a very important “IF”. Certainly a 3.84 at Chicago makes the point that you are academically qualified to handle the course load at Columbia. This point will matter if you ever get to the point of applying to a graduate program at Columbia and if your Chicago GPA is still that good.