If there is anything else introduced into equations like the complex numbers. I am trying to understand what complex numbers really are; they seem more of a syntactic feature of equations than anything else, but I'm not sure. As such, though, I am wondering if there are any other examples throughout Mathematics of these sorts of "hoops" that are being jumped through, so to speak, mathematically. I am not talking about extensions of complex numbers like quaternions and such, but more along the lines of things perhaps that were added into the mix of equations to help solve them, though they aren't really "real" or they have this quirkiness like the complex numbers do.
 A: Basically all of the numbers which we use are nearly incomprehensible abstractions.  Frankly, I don't believe that any number other than 1, 2, or 3 is really real.  It is a testament to the power of human cognition that we have invented mathematical structures which generalize these three numbers so very consistently.
The Natural Numbers
I have made the bold claim that 1, 2, and 3 are the only really real numbers.  I mean, if I have 1 apple, then get another apple, I have 2 apples.  Add another, and I have 3 apples.  Add one more and I have... many.  I have many apples.  Please don't give me another apple.  I'm full.
This version of counting might be good enough for many purposes, but doesn't really cut the mustard in general.  So we create the natural numbers, which capture this notion of successors.  Given a natural number, there is always a "next" number.  These numbers are perfect for counting objects, and solve the problem of keeping track of my apples.  Moreover, natural numbers allow for basic accounting—I can add and subtract quantities, for example.
These numbers are totally fake.  Like, a complete abstraction.  I don't believe in them.  But they are useful.
Zero
Okay, fine.  I might not like it, but I accept the naturals.  They're groovy.  I can kind of wrap my head around them.  But they still don't do everything I need.  For example, if I have 5 apples, but if 5 raccoons get into my house and eat each eat an apple, I can't tell you how many apples I have.  I clearly have no apples, but "no" isn't a number.
Except, I suppose, that it is is a number.  Zero.  I just invented zero!  I am so cool!
Zero was actually something of a radical idea in mathematics.  The modern zero comes to us by way of Islamic scholars, with the original idea coming out of India.  As Servaes pointed out in a comment, it took a while for zero to be acceptable in Europe.  Zero was also independently described by some indigenous American groups (though those notions don't really influence the modern understanding).
Negative Numbers
Suppose that I have an apple orchard, and you have some sheep.  In the spring, you shear your sheep.  I need some wool, so I come to you.
You say "Great, I have wool.  I need 10 bushels of apples."
This is a problem.  My apples aren't going to be ripe until the fall, so I ain't go no apples right now.  So we need to come up with a way to get me some wool while keeping track of the apples that I owe you.  Essentially, we are saying that, while I have zero apples right now, I owe you 10 bushels of apples.
This gives us the negative numbers.
The negative numbers are garbage numbers.  They make absolutely no sense at all.  I am perfectly willing to agree that if I have "no" apples, we can consider that a number of apples, but negative apples?!  Nonsense.
On the other hand, negative numbers are an incredibly useful tool for accounting.  So, I guess I'll have to put up with them.  Sigh.
Rational Numbers
One of the ways that I add value to my apples is to make pies, which I can sell or barter away.  So, I make some pies, and you come along wanting a slice.  Each pie costs 12 bucks, but you only have $2.  I guess you can't have any pie.
However, you are pretty clever, so when you ask for a knife, I give loan one to you.  I'm a little suspicious—I really don't want to get mugged by you—but you have an honest face.  You cut my pie up into six pieces, then give me your $2 in exchange for a piece of pie.  How much pie did you buy?
Well, one pie gets divided into six pieces, and you took one of them.  That is, you took 1 of the 6 slices or, as we would say today, $1/6$-th of a pie.  This gives a basic idea of rational numbers, but are good for dividing things up.
Historically, the Greeks worked with rationals.  They understood a number to represent a length.  The number $\frac{2}{3}$ was a stick that, if copied three times and laid end to end, would be the same length as two sticks of some predetermined "unit" length.
Right.  Okay.  Fine!  There are rational numbers.  Now what?
The Real Numbers
It seems like the rational numbers ought to be good enough for anything.  It isn't too hard to show that the rational numbers are dense, in the sense that we can always find a third rational number between any two given rational numbers.  This means that if we are given any length at all, we can find a rational number that is a little bigger, and a rational number which is just a little smaller.  This is actually sufficient for any kind of computation that we would ever want to actually carry out, and is what computers do (what is a floating point number, other than a rational number with a moderately large denominator?).
Unfortunately, we can easily show that there are things which ought to be numbers, but which are not rational numbers.  For example, my father started our family apple farm.  The orchard itself is a perfect square, with a juicing plant at one corner, and a saucery at the opposite corner.  When dad died, my sister and I needed to split the orchard between us and both have access to the facilities, so we split the orchard right down the diagonal.  However, my sister and I hate each others guts, so I want to build a wall.  How long is that wall going to be?
It turns out that the length of the wall is going be related some some number which, when squared, is equal to 2.  But no such number exists (because the only numbers which matter are rational numbers).  Just ask Hippasus.  In order to get around this, we invent irrational numbers.  Together with the rationals, that gives us the reals.
The key property here is completeness.  The rational numbers are not complete in the sense that there are sequences of rational numbers which ought to converge, but which don't converge to rational numbers.  A sequence of decimal approximations to $\sqrt{2}$ is a good example.  The sequence
$$1, 1.4, 1.41, 1.414, 1.4142, \dotsc $$
has the property that terms very far along in the sequence are close together (this property is the Cauchy property), yet the sequence fails to converge in the rationals.  This will never be true in the reals.  Any sequence of real numbers which has the Cauchy property will converge to a real number.
This might not seem like a big deal, but it is the heart of calculus and analysis.  Calculus and analysis are the tools which we use to calculate orbits of satellites or model the vibration of a string.  We can perform computations with only the rationals, but we can't actually prove that those computations make sense without the reals.
But the reals are totally made up.  Fake news.  They don't exist.  In fact, you can't even write down most of the real numbers!  They just aren't right!
The Complex Numbers
At some point along the way, mathematicians became interested in polynomials.  The quadratic formula was known in antiquity, but it was understood that if the discriminant $\Delta = b^2-4ac$ was negative, then the corresponding quadratic polynomial had no roots.  This was fine until Tartaglia and Cardono came along and started cranking out the roots of higher degree polynomials.  These crazy guys actually treated the square roots of negative numbers as though they made sense!  Double ewe tea eff!?
These computations were utter nonsense, of course, because the negative numbers don't have square roots.  On the other hand, if you accept the square roots of negative numbers, then you get the complex number system.  The complex numbers are useful, because they are algebraically closed.  That is, if you have a polynomial with real (or even complex) coefficients, it will have all the roots it should within the complex number system.

TL;DR
The moral of the story is that the history of mathematics is full of "complete" number systems which ultimately don't solve all of the problems which one might want to solve.  Humans have been inventing new kinds of numbers for thousands of years in order to get around this.  Negative numbers, zero, the rationals, and real numbers are just as weird and quirky as imaginary numbers.  We are just exposed to all of these other number systems much earlier in our lives, and are therefore more willing to accept them.

Suggested Reading
In terms of references, there are a couple of things you might want to read:

*

*Lakoff, George; Núñez, Rafael E., Where mathematics comes from. How the embodied mind brings mathematics into being, New York, NY: Basic Books. xvii, 493 p. (2000). ZBL0987.00003.
Lakoff and Núñez (or my poor recollection of their work) is the basic model for the story I outlined above.  They are cognitive scientists interested in the way that the human brain understands the world.  There mathematics is not always right on, but I think that there is insight to be gained from their work.


*Boyer, Carl B., A history of mathematics. Revised by Uta C. Merzbach., New York etc.: John Wiley & Sons. xviii, 762 p. £ 14.95 (1989). ZBL0698.01001.
Honestly, any history of mathematics should suffice, but I consider this one to be fairly authoritative.  This should give some historical perspective on the development of ideas.
A: p-adic numbers is my suggestion:
In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.
Lifted from Wikipedia : https://en.wikipedia.org/wiki/P-adic_number
