Whatever value x you assign to 1/0, it had better be true that multiplying it by 0 gives 1; and the properties of a field give that 0x = (0 + 0)x = 0x + 0x, so that 0x = 0. You thus would have to give up either distributivity or additive inverses, which are pretty dear to me!
If, on the other hand, you regard y = 0/0 as a "multi-valued variable" standing for any element of the field, then it behaves perfectly fine, since 0y = 0. The only problem is that it's infectious, so that just about any arithmetic computation involving it, like y + 1 or 2y, also suddenly has to stand for every element of your field. (If you work over a ring, then y + 1 stands for any element and 2y only stands for elements of the ideal generated by 2, etc.) Computations like 0y and y + (-y) both still yield 0.
This is not very useful—at least I can't see anything useful to come of it—but, aside from the infectious multi-valuedness, I don't see what problems result.
If, on the other hand, you regard y = 0/0 as a "multi-valued variable" standing for any element of the field, then it behaves perfectly fine, since 0y = 0. The only problem is that it's infectious, so that just about any arithmetic computation involving it, like y + 1 or 2y, also suddenly has to stand for every element of your field. (If you work over a ring, then y + 1 stands for any element and 2y only stands for elements of the ideal generated by 2, etc.) Computations like 0y and y + (-y) both still yield 0.
This is not very useful—at least I can't see anything useful to come of it—but, aside from the infectious multi-valuedness, I don't see what problems result.