My mother got seriously burned when I was young. She used an ointment made from calendula. It was homeopathic... though my family is sceptical about homeopathy in most cases. I don't remember any dark scars, and do remember lots of good comments about the ointment.
Can't help with home remedies but I was burnt around the face, ears and neck must over 8 years now. The advise was I given was to use Bio Oil (available at chemists although I think Countdown stocked it for a while) continually for over 6 months, although the A & E gave me a black tube colour of cream esp. for burns immediately after the incident.
The only scars I have now is where my necklace was burning against my skin and stretched skin around my neck (looks like a poor facelift job). But the main thing is to keep applying the oil. I was also warned in summer my face would be patchy and to keep the sun off of it.