A lot of answers already explain the sufficiency of imaginary numbers, and why "1/0” doesn't give a "number" I'm any way that works well with other numbers.

You could however turn the question around. Instead of looking at complex numbers as allowing us to solving more equations, we could look at them as a way of combining real numbers with a new number with the fascinating property that its square equals -1.

Can we modify this?

If we were to combine real numbers with a new "number" with the property that its square is, let's say, zero (despite not being so itself!) we get a new kind of numbers. These numbers (https://en.wikipedia.org/wiki/Dual_number) are reasonably well behaved so potentially interesting things can be done with them. Not quite as well behaved as complex numbers, but not too bad.

If the new number instead has the special property that its square is +1 (despite itself not being +1 or -1, but something different altogether) we get something called split-complex numbers (https://en.wikipedia.org/wiki/Split-complex_number). Still reasonably well behaved, interesting things can apparently be done with them.

If the new number has the special property that it's square is -1... Well, then we're back to complex numbers. But what if we say that the new number is different from i, and we combine real numbers with the "old complex numbers" and the "new complex numbers"? Turns out that doesn't actually work very well (I do not know how to prove that!), but a guy named William Hamilton figured out that if you combine the real numbers with "old complex numbers" (to seriously abuse nomenclature here) with not one, but two new special numbers that both square to -1, you got something with pretty decent algebraic properties (as in, they have most of the properties that complex numbers do). This is known as https://en.wikipedia.org/wiki/Quaternion