Would $1/0$ be accepted as a unit if it were useful? My layperson's impression is that $i$ (i.e., $\sqrt{-1}$) is accepted as  unit in mathematics not because it is meaningful, but because it is useful: you can use it to solve polynomials; you get Euler's formula out of it; lots of stuff.
Both $\sqrt{-1}$ and $\frac{1}{0}$ are similarly nonsensical concepts to me (as a mathematical layperson). But suppose someone came up with an compelling and fascinating results that follow, if you accept that there's some number $k = \frac{1}{0}$. Would mathematicians just forget that $\frac{1}{0}$ is strange concept, and start using $k$ in their work?
 A: The difference between the two examples is that a solution to $x^2+1=0$ is not inconsistent with the axioms of a field, a solution to $0x=1$ is inconsistent with those axioms.
Now, a solution to $x^2+1=0$ is inconsistent with the axioms for an ordered field, so one could make a case that they are equally infeasible in that context.  Perhaps historically this was true at some point: fields grew out of studies of the rationals, reals, and ordered geometry, if Hilbert's Grundlagen der Geometrie is any indication. I think if they were assumed to be ordered, it probably even predated the term "field."
But that's not the case today: we let fields be unordered so that $\mathbb C$ is one of our playthings. We were OK with giving up "ordered" from our axioms.
What does one have to give up to accept a multiplicative inverse to $0$? Quite a lot. The problem with $0x=1$ is that if you are assuming your algebra is distributive, you necessarily have $0x=0$, and that would mean $0=1$, which is not possible given our definition of what a field is.  If you give up $0\neq 1$, then you can have $0x=1$, it's just not very interesting because your object can only have one element, namely $0$.   You could also give up an identity entirely, but then the whole notion of division and units is lost.   You could also give up distributivity, but then you have no relationship between addition and multiplication, and most of the useful algebra vaporizes.  This is all quite a lot more to overcome than simply dropping an order axiom.
Finally, one could also stop asking for the two operations to obey rules on the entirety of the set (so that you could exempt $1/0$ from satisfying certain axioms).  While a little awkward, I think this is the most common thing that happens, and it is not so unpleasant given the way it cooperates in classical cases.
I'm not saying $1/0$ is completely unacceptable, it's unacceptable in the normal theory of fields.  The fact that topologically the "shape" of something like $\mathbb C$, when operated on by linear fractional transformations, suggests one can successfully fill a vacancy with a new character, and furthermore that character cooperated with the linear fractional transformations, is fantastic.
To me that means that $1/0$ is at least "topologically acceptable" or maybe "analytically acceptable," but some may take issue with the accuracy of those descriptions.
So, you can see there is a little grey area here. In summary, I think the movement to accept the existence of $i$ was not as messy.
