A proposal that shows up every so often is to use dozenal arithmetic. The idea is simple, in base 12 more fractions work out very conveniently. You can easily divide things 3 and 4 ways. This is frequently convenient, and is why we often sell things in dozens. It is also why the much maligned Imperial system sneaks factors of 3 (3 tsp in a tbsp) and 12 (inches in a foot) in various places. And there are numerous minor benefits, such as the fact that the multiplication table becomes significantly simpler and therefore easier to learn.
When the French created the metric system, they based it on factors of 10 everywhere. They even went so far as to try to measure angles in gradians (a quarter circle had 100), and to use decimal time. The world as a whole rejected decimal angles and times, but has adopted decimal metric everywhere else. Which is very convenient for scientists, but is hard to divide into thirds and somewhat inconvenient for quarters. Furthermore the persistence of both systems results in occasional annoyances like the fact that in daily life we usually prefer to measure speed in km/h, but for energy and power calculations the units only work out properly if you measure in m/s. Resulting in an annoying factor of 3.6 that shows up converting between them.
The other day I read An Argument for Dozenalism that made many of these arguments. Nothing new. However it made the interesting point that ideally a dozenal arithmetic would have its own set of glyphs. It suggested that 0 and 1 could be kept, but everything else should be changed. In my opinion that argument is correct, it is very confusing if 21 sometimes is 5*5 and sometimes 3*7, which makes a mixed dozenal and decimal world harder than it needs to be. But this raises the interesting question of what a logical dozenal set of glyphs might look like.
I've amused myself with thinking about this, and I have a proposal for a set of glyphs that are (mostly) unused, easy to learn, quick to draw, and are much more logical than existing ones. All have a vertical line in the middle. At the top there is a choice of a hook starting on the left, a straight end, or a hook starting on the right. At the bottom there is a choice of a hook ending on the left, straight down, on the right, or bending right to cut across the vertical line. This gives 12 possibilities. By incrementing the bottom first, and the top when you get a carry, you get a sort of a 3,4 base. Which means that a glance at the glyph tells you immediately its sign mod 4. A glance at the top tells you whether it falls in the range 0-3, 4-7 or 8-11.
So instead of the current system of 10 random symbols to memorize, you get two simple rules. While at first it seems bizarre, it grows on you quickly. And it gives you a very quick way to tell whether a given number is written in decimal or dozenal. To get a sense what it looks like, take a look at the 12 times table (my apologies for the handwriting - my none too good penmanship gets worse when I'm using a mouse pad to draw with):
Note in particular how regular the patterns are for 2, 3, 4, and 6. Doesn't this look easier to memorize than the decimal times table? As an exercise try writing out your favorite sequences. Whether you're writing out squares, powers of 2, or primes you'll see that more patterns leap out at you in dozenal, making them easier to learn.
So there is my humble contribution to a dozenal future.
(In other news, I now have a patent to my name, though in fact it is owned by a previous employer. By the standards of the patent system, it is not a particularly bad patent. But if I had my druthers, it would have never been filed. Ah well.)