Before we talk about complex numbers, let's make sure we understand real numbers. Real numbers, basically, are the numbers you see on a number line. It's your numbers like -2, -1, 0, 1, 2, and everything in between. This includes both rational and irrational numbers. The square of a real number is always nonnegative, so we know that, if x2=-1, x is not a real number.
Now, think about a point (a, b). It's just a pair of real numbers. It can be drawn in a plane using x and y axes. This is essentially what a complex number is: a point in a plane. But, instead of (a, b), we write a+bi. Also, we can do addition, subtraction, multiplication, and division with these objects. Just as you might write
(a, b) + (c, d) = (a+c, b+d)
for a vector, we write
(a+bi) + (c+di) = (a+c) + (b+d)i.
Multiplication might be where it gets confusing. Let's first think about how multiplication works for real numbers. A positive number times a positive number is a positive number. A positive number times a negative number is a negative number. A negative number times a negative number is a positive number. We can think about multiplication by a number as scaling: multiplying three by two is to scale the distance of three to the origin (zero) by a factor of two, moving it to six. Multiplying three by negative two not only scales the distance to the origin by a factor of two, it also flips it 180 degrees around the origin, landing it at negative six.
Complex numbers work the same way: when you multiply 3+4i by 2, you obtain 6+8i, since it scales it by a factor of 2. When you multiply 3+4i by -2, you obtain -6-8i, which is twice the distance to the origin that 3+4i was, but is also pointing in the opposite direction. It gets slightly more complicated when you're multiplying a complex number by a nonreal complex number. Let's think about multiplication by i. Since the distance of i to the origin is 1, it shouldn't increase the distance of complex numbers to the origin. However, from the number 1, the number i is rotated by 90 degrees. Actually, multiplication by i rotates complex numbers counterclockwise by 90 degrees, as if they were vectors! For example, (3+4i)*i = -4+3i. Draw it on the plane to check it out!
This is actually what complex multiplication is. Take two numbers, a+bi and c+di. Check out their angles with the positive real axis (the x-axis). Add them. This is the angle with the positive real axis the product of the two numbers will make. Next, check out their distances to the origin (using the Pythagorean theorem). Scale one by the other. Now, you've got an angle and a magnitude, uniquely describing a new complex number! Namely, (a+bi)(c+di) = (ac-bd) + (ad+bc)i.
So, the next time you see the complex number i, think of it as not just the square root of -1, but as a counterclockwise 90 degree rotation! After all, performing a counterclockwise 90 degree rotation twice gets you at a 180 degree rotation, which is multiplication by -1. It becomes clear why multiplying a complex number a+bi by its conjugate a-bi results in a real number, since, by being symmetric about the real axis, their angles with it add to zero! Thinking of complex numbers in this way provides a lot of good intuition for them.
