The sum of two complex numbers is always a complex number?

A "complex number" is a number of the form a+bi, where a and b are both real numbers and i is the principal square root of -1.

Since b can be equal to 0, you see that the real numbers are a subset of the complex numbers. Similarly, since a can be zero, the imaginary numbers are a subset of the complex numbers.

So let's take two complex numbers: a+bi and c+di (where a, b, c, and d are real). We add them together and we get:
(a+c) + (b+d)i
The sum of two real numbers is always real, so a+c is a real number and b+d is a real number, so the sum of two complex numbers is a complex number.

What you may really be wondering is whether the sum of two non-real complex numbers can ever be a real number. The answer is yes:
(3+2i) + (5-2i) = 8.

In fact, the complex numbers form an algebraic field. The sum, difference, product, and quotient of any two complex numbers (except division by 0) is a complex number (keeping in mind the special case that both real and imaginary numbers are a subset of the complex numbers).