Let's assume such a number exists...call it a. How far can we get? What rules of division and multiplication do we need to meet?

That would be quite impractical. If you're to work it out, changing the division in real numbers in such a way would create two absorbant elements in real mulitplication, which would then lead to having to reformulate many of the other division properties. That would also lead to restating what a positive and negative number would be, which would then make you...get my point ?

You can go ahead and define such a number "a", but unless you find practical use for it, and allow it to coincide with the other properties defined in regular arithmetic, I doubt it will be wide-spread and accepted.

Even at that, I doubt a ring will ever exist in which division by 0 is defined, simply due to the fact that no one needs to know what a number divided into 0 distinct entities is equal to. The same cannot be said about negative roots and logarithms, whose necessity were the prime motive for the creation of complex numbers.

When they find a motive for 3/0 to have a result, then the appropriate result will be given. Until then, it will be undefined.