This is where advanced algebra really starts, this topic typically comes after quadratics and polynomails. Complex numbers give you another dimension of math. Complex numbers are a formation of real numbers (the numbers you're used to, such as 1, 3, 5, -4 etc) and imaginary numbers.
Now, you may be wondering what imaginary numbers really are and how they came about. Think of how we get a root function and how it works: you feed the function an input and it gives you a number out that descirbes the number in terms of a number and its index. For example if we have a number n squared, and we take the sqaure root (index = 2) of n squared, we get n, since it is n x n. If we take the sqaure root of 16, we get 4, since 16 is the product of 4 x 4. Try doing that operation with negative numbers. It does not work! And this is where imaginary numbers stemmed from; the roots of negative numbers. Imaginary units are descirbed in terms of a number i, the (imaginary) number i is defined as the square root of negative one.
Now that we understand what complex numbers are, we can go on. So, as I was saying, Complex numbers are made of real numbers and imaginary numbers (more like imaginary parts). The usual notation is 
. Z is any complex number (It's the variable most used to conotate complex numbers) a is the real part, b is defined as the imaginary part, and i is just the imaginary unit. One mistake very common is that people think the "imaginary part" is bi, that is wrong. The imaginary part is only b.
If I have a complex number z = 5 + 3i, then Im(z) = 3 and Re(z) = 5.
Where L is the set of imaginary numbers and R is the set of real numbers. This just means "C is a superset of R and L". Which in layman's terms means set C (set of complex numbers) is a set which contains set R (Set of real numbers) and set L (Set of imaginary numbers). This is useful to know because Complex numbers are really the biggest sets we know of so far, maybe there is something bigger and more abstract, but for practical reasons, this is the biggest set in mathematics. You probably have heard that complex numbers can be used as vector representations. That is indeed true, since the imaginary number line is an extension of the real number line, they create two axes which resembel the conventional x and y axis. This is called the Complex plane.
So if I have a complex number z = 4 + 3i, I could think about it as (Re, Im), since we have the Real axis and the Imaginary axis. So in this case we could plot it on the complex plane as (4,3). NOTE: 4 + 3i and 4 + i3 are the same number.
To find the magnitude of the vector z we use a modulus operator which is defined as, for a complex number z = a + bi, as 
. So for the complex number (4 + 3i) in the example above, we know that | z | = 5.
In Complex number theory there exists a function called the argument of a complex number, notated as arg(x+iy). The argument of a function is the same as arctan(y/x). This function obtains the angle between the horizontal axis and the complex-number-vector.