Apparently Carmack didn't actually use it anyway, so I suppose there wasn't any purpose. But Carmack's series is significantly different from the normal Fibonacci series, because it only consists of the first ten elements, which isn't enough time to converge. If it was used in actual graphics or gameplay, it would have had a significantly different effect than a normal Fib sequence.
I just thought it was interesting, and another example of the same type of style which led to rocket jumping or bunnyhopping, which were also accidents.
It's not clear that Carmack's purpose in using the Fibonacci series was to take successive ratios to approximate "the golden ratio" (if it was the golden ratio he cared about, he could just as well have directly calculated (1 + sqrt(5))/2). He quite possibly could have been interested in the Fibonacci sequence for some other reason, or no reason at all.
I just thought it was interesting, and another example of the same type of style which led to rocket jumping or bunnyhopping, which were also accidents.