Could 1/0 be an imaginary number? There is no way to find the square root of a negative number. It just doesn't work. So the answer to the impossible question, "What number squared equals a negative number?" is just said to be $i$, an imaginary number. 
So now let's look at a different problem. What is one divided by zero? Of course, you can't answer that question. It doesn't make any sense. Splitting up a chocolate bar to a group of 0 friends isn't possible. It's like being told to walk north when standing on the north pole. So, what if just like $i$, we just say $1/0$ is an imaginary number, referenced by the letter $o$?
What applications would this have? Does it even work? The number $i$ has real life applications, but it can also be used to create abstract designs like the Mandelbrot Set. Could $o$ even be used for an abstract purpose?
 A: Fundamentally, complex numbers were used because they arose naturally in polynomial roots. However, division by zero has no such purpose. In fact, if division by zero was defined, then $(0\cdot 1)\frac 10=(0\cdot 2) \frac{1}{0}\implies 1 = 2.$ (Oops). However, if you are willing to forego standard properties, then you get an algebra called a wheel. You can learn more about it here: https://en.wikipedia.org/wiki/Wheel_theory
A: "The number $i$ doesn't have many real life applications" is a very false statement. Indeed, much of electrical engineering relies on complex numbers. The solution to differential equations (both ordinary and partial) also often involves complex numbers, of which we find applications for in nearly every area of science and engineering. 
That's besides the point though. It's an interesting thought you raise. It seems that we get some very nice structure (quite beautiful to be honest) and interesting properties out of this set that we call the complex numbers. Naturally, you might be compelled to ask - or even expect - to get the same thing out of the other famous "undefined operation' - division by zero. As it turns out, however, there's no number system in which division by zero is defined. The process breaks down almost immediately. There's a quite popular area that deals with stuff like this called "mathematical fallacy" (see https://en.wikipedia.org/wiki/Mathematical_fallacy), but here's a quick way of showing why division by zero leads to nonsensical answers:
$$\text{Let } a=1=b$$
$$\Rightarrow a^2=ab$$
$$\Rightarrow a^2 - b^2 = ab - b^2$$
$$\Rightarrow (a-b)(a+b) = b(a-b)$$
$$\Rightarrow a+b = b \text{ (observe that we divided by } (a-b)=(1-1)=0 \text{)}$$
$$\Rightarrow 1 + 1 = 1$$
Once you have this it becomes hard to make a case for any kind of benefit for defining such an operation. It should also be noted that in more serious mathematics, the complex numbers tend to arise quite naturally, with the definition $\sqrt{-1}=i$ being completely done away with. This post is already quite long, but here's a starting point for research if you want an example of how to do this: Can the complex numbers be realized as a quotient ring?. Hope that helps!
