I have been reading up on boolean algebra quite recently, for those not familiar, this type of mathematical system has much to do with the way logic is represented (and is primarily applied to, though not limited to circuits). I have learned that the result of an equation, even something as simple as 1+1, is different compared to traditional mathematics. In Boolean logic, 1+1 = True, obviously in traditional maths it's 2.
Though what I really find interesting is not the differences which boolean algebra and traditional algebra share, but their similarities.
For instance, I have found that the relationship between the two algebraic systems are not just similar, but identical, even though the product is different. This becomes clear when we replace the numbers themselves with variables that reperesent these numbers, but simutanously shines light on their relationship as well. So, AB = BA, A+B = B+A. A(B+C) = AB+AC, and so on. This is true in both systems, even though the product is very different. For those familiar with boolean algebra, you will know that the only two possible answers are 0 and 1.
So then, it seems that the results can differ, yet the relationship within a system does not, though is this actually true? Is there such a system where A+B = AB + AC, or perhaps AC = A^2+C. Don't get me wrong, this seems absurd, and would perhaps be at the boundaries (or perhaps beyond) our intellectual limits, but that's the point, such a system would be very unintuitive, but would be otherwise correct, that is if it actually represented certain properties of the universe.
Certainly such things are possible. The question is: are they interesting?
For a while it can be fun to play the "symbols game", where you invent meaningless objects and play around with them. I did a fair bit of this a few years ago, and I don't regret the effort. But if you ever want anyone else to be excited about what you're excited about, then you will have to either (a) come to some objectively interesting conclusions (by which I mean, having more merit than simply unintuitive consistency), or, more likely, (b) relate your symbols to other symbols that other people care about for historical reasons.
I'm not aware of any serious attempt to relate these sorts of ideas to either the "mainstream" mathematical or the known physical universes, although as Slade mentions, universal algebra is probably the closest thing around.
For a (very rudimentary) starting point at playing the symbols game, I'd recommend the book Negative math, which is an attempt to take seriously that age-old middle-school question "Shouldn't minus times minus equal minus?". IMO, the book thinks a little too much of itself, but if you can get over a bit of pretension then you probably would enjoy the ride.