I finally got around improving and adding a few things on what I had.
This is the current version of the fire.
I tried to give it more pink-ish colors instead of the classical yellow-red range.
Added a few sparkles at the base and that fly away on top.
There is some very basic refraction at the base as well to deform the light on the sparkles.
Finally I added a “heart” in the center of the fire with a very high emissive and vertex offset that gives a glow that flickers and some substance to the whole thing.