How would you explain a quadratic field to a beginner? How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more concrete terms.
Please also give a example question and answer.
For example, a college sophomore like me who knows things through calculus, high school algebra, and some modular arithmetic.
 A: Oftentimes in mathematics we find that by expanding our universe we can find out more about a subject that has its roots in something simpler. Look at polynomials over the real numbers, some of them don't have roots, but all of them have roots in $\Bbb C$, the complex numbers. Quadratic fields grew out of the early study of quadratic forms in two variables, i.e. things of the form
$$ax^2+bxy+cy^2$$
the case $a=c=1, b=0$ gives $x^2+y^2$ which is exactly the so-called "norm" function on $\Bbb Q(i)$, i.e. for $x,y\in\Bbb Q$ we have
$$N(x+iy)=x^2+y^2.$$
And since we're number theorists, we care mostly about integer solutions to things like $x^2+y^2=n$. After expanding from just "integers whose square sum equals another integer" to "elements of $\Bbb Q(i)$ with integer coordinates so that their norm is $n$" we find that the latter question is easier to study because things like $N(\alpha\beta)=N(\alpha)N(\beta)$, and because you can stick $\Bbb Z[i]$ into $\Bbb C$ in an obvious way and thanks to the Gauß circle problem, we can use this philosophy to even count about how many ways we can find solutions!
So the purpose of dealing with the field is that it has a lot more useful structure for us to build on there, and it is a lot easier to work with than working directly with the integers. A similar problem that is solved in this fashion from my original example of polynomials over $\Bbb R$, there is a theorem that the roots of real polynomials come in complex conjugate pairs with equal multiplicities, the easiest way to prove this is to


*

*Factor the polynomial over $\Bbb C$ as


$$c\prod_{i=1}^n (x-r_i)$$


*

*Apply complex conjugation to this expression


But you don't get this tangible, computable expression without the factorization into linear factors, which is only necessarily true over $\Bbb C$.
Other quadratic fields are used to study other quadratic forms, and so we developed a great deal of theory for them for this reason, although--in the interim--they have taken on a huge and rich theory of their own! One example, which is perhaps a bit more advanced than where you are, but not too far removed (since you've studied modular arithmetic) is that the information we can glean from some quadratic fields, can tell us about solutions to $ax^2+bx+c=0$ over the integers modulo $n$, that is:  it can help us figure out when there are solutions to quadratic congruences via the law of quadratic reciprocity.
A: I'll start with the disclaimer that I've never been a math teacher and I'd probably be awful at it.
First I'd want to know whether the beginner's interest in mathematics is more algebraic or more geometric.
By algebraic, I'm talking about solving equations like $x^2 + 3y^2 = 5$ or $x^2 - 47y^2 = 37$. An example question would be whether these equations have solutions in integers. The answer is that the first equation does not, while the latter has infinitely many (e.g., $x = 15$, $y = 2$). Surprisingly, determining the former has no solutions can make use of imaginary numbers. You can try solving equations like these by just trying a whole bunch of values, which might not be so bad in the plus case, but can easily be trying in the minus case even if there are infinitely many solutions.
For someone with more geometric interests, I would go ahead and introduce the concept of the complex plane and numbers of the form $a + b \sqrt{-d}$, where $d$ is a positive integer and $a$ and $b$ are any integers (including $0$). You can't really do greater than or less than on these numbers, but you can compare how far these numbers are from $0$. An example question would be which of $-7 + 5 \sqrt{-2}$ and $12 - \sqrt{-2}$ is closer to $0$, and what part the Pythagorean theorem plays in this.
For someone more interested in the historical aspect of it, I suppose it comes down to the romance of Fermat's last theorem, "a simple knot tied by a part-time French mathematician working alone without a computer" and not untied quite to the satisfaction of a French space captain in the age of truly ubiquitous computers.
There is also the number theoretic perspective, for questions such as what prime numbers can be solutions to an equation like $x^2 - 10y^2$. It's not for nothing that there is an entire book titled Fermat's Last Theorem and Algebraic Number Theory.
