When I read your answer, I was thinking, (seriously no offense because I know you are really smart) I don't know for sure that this definition works for complex numbers.
It does; complex numbers are just another type of number. We'll get to them shortly.
And then I was thinking that mathematics relies on definitions and deductive reasoning and intuition cannot give the certainty of deductive reasoning, thus it might be a fallacy to think that something simple and intuitive is an accurate model of mathematical reality... then I remembered that it was taught in kindergartens even...
To be fair, sometimes the intuitive answer is wrong; one does have to take care. But sometimes, as in these cases, the intuitive model does work.
Define "multiplication"
X*Y : I have Y sets of X objects, how many objects do I have?
Exactly.
"addition"
X+Y : I have X objects. I am given Y objects. How many objects do I have?
Perfect.
It's easy to visualize imaginary numbers as another type of object 'x', and I am given y objects. So I have x + y imaginary objects and X + Y real objects.
You could do it that way, and it leads to the correct answers, but I think it's fundamentally problematic to see complex numbers as intrinsically different to real numbers. (For one thing, real numbers are a subset of complex numbers in any case).
"subtraction"
X-Y : I have X objects. Y objects are taken away from me.
Right.
Then it makes me wonder what other exceptions to manipulation there is
There's only one that I can think of off the top of my head; if x^z=y^z, this does not mean that x=y (i.e. we can't just take the z'th root on both sides of the equation). This can be clearly demonstrated with x=2, y=-2 and z=2. Two squared is four, which is equal to (negative two) squared, but two is not equal to negative two.
Now, as to complex numbers. Let me start by asking you to define a "complex number".
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