Currently, we use komi (bonus points for White) to try to resolve the inherent unfairness of the BWBWBWBWBWBW order of play, clearly favouring Black. There is much debate about what value best achieves this and whether the same value is true for all 3 standard board sizes.
In the past, before komi, some people would play a jubango (a 10 game match) where players would take it in turns of who got to play black and white, but this too followed the ABABABAB pattern (unless one player won enough games with the weaker side that his opponent was forced to take the handicap of always having the stronger stones). You also need the time to play your opponent 10 times.
Mathematics has its own approach that is not dependent on set (board) size. The Thue-Morse sequence. Instead of ABAB it starts ABBA, immediately the advantage of going first is parried with the advantage of going twice. To continue, you add the inverse of the whole and you get ABBABAAB (and as you reach the end of a sequence, just inverse the whole and continue on, with the next stage looking like ABBABAABBAABABBA). This sequence tends towards total fairness as n increases, so given that most games go well over 100 moves, by a game’s natural end there will be no advantage to either white or black.
Not only would this solve the problem of komi, but it would also add in some exciting intricacies for tesuji, joseki, fuseki, shape, and ko fights depending on the timing of when each player gets their double turn.
Obviously this would result in the game looking almost nothing like how we currently play (due to it impacting essentially every element of the game) but I think it would be something fun to play around with! (preferably online with turn order programmed in so that you don’t have the headache of working it out manually)
Further reading on Thue-Morse