Really, I could say that numbers themselves don't exist. Imagine I am an alien from the planet Krypton who just came here knowing nothing about numbers. All I know that are real are trees and apples and popcorn.
"Numbers?!", I might say. "Well if I can't see them, how do I know they are real?" Then you might get three apples and say to me, "Look here, see these apples?" I would nod my head, and you might say, "Well, numbers basically show us how much of something there is. I will explain the words we use to describe numbers." Then you may put away two of the apples, leaving one in your hand. Then you would say, "Here is one apple." Then you would get another apple and say, "Here are two apples." You would get the last one and say, "Three apples". Then you would start teaching me more numbers.
*Fast forward 15 years later*
Now I come to you again, asking you, "What are complex numbers?" How would you explain that to me? You might say, "Remember when I taught you what numbers were? It was like learning a totally new language! From there you learnt many things about numbers, which we humans call Mathematics. You learnt about algebra, which is just another part of the language of mathematics. You learnt many, many things about math. Now I am going to teach you about complex numbers. Think of it as yet another part of the language of mathematics."
"Remember when I taught you about the natural numbers ($\mathbb N$), then integers ($\mathbb Z$), then rational numbers ($\mathbb Q$), them real numbers ($\mathbb R$)? $\mathbb Z$ closed off $\mathbb N$ with negative numbers, $\mathbb Q$ closed off $\mathbb Z$ with fractions and decimals, and $\mathbb R$ closed off $\mathbb Q$ with irrational numbers like $\pi$. What then, closes off $\mathbb R$? Complex numbers ($\mathbb C$) do." After explaining what complex numbers are, you would say, "See how complex numbers close off the real numbers and fill in all the remaining gaps? If only real numbers exist, we would never be able to explore negative square roots. But with complex numbers, we can!" Now I might ask, "What closes off $\mathbb C$ then?" You would reply, "No one knows. Maybe one day, someone will invent a new number system that closes off $\mathbb C$, but as of now, there is nothing that closes off $\mathbb C$."
That is what I have to offer on the subject of complex numbers.