Is there a problem in defining a complex number by $ z = x+iy$? The field $\mathbb{C} $ of complex numbers is well-defined by the Hamilton axioms of addition and product between complex numbers, i.e., a complex number $z$ is a ordered pair of real numbers $(x,y)$, which satisfy the following operations $+$ and $\cdot$:
$ (x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1+x_2) $
$(x_1,y_1)(x_2,y_2) = (x_1x_2-y_1y_2,x_1y_2 + x_2y_1)$
The other field properties follow from them.
My question is: Is there a problem in defining complex number simply by $z = x+iy$, where $i² = -1$ and $x$, $y$ real numbers and import from $\mathbb{R} $ the operations? Or is this just an elegant manner to write the same thing?

There is no "explicit" problem, but if you are going to define them as formal symbols, then you need to distinguish between the + in the symbol $a$+$bi$, the $+$ operation from $\mathbb{R}$, and the sum operation that you will be defining later until you show that they can be "confused"/identified with one another.
That is, you define $\mathbb{C}$ to be the set of all symbols of the form $a$+$bi$ with $a,b\in\mathbb{R}$. Then you define an addition $\oplus$ and a multiplication $\otimes$ by the rules
$(a$+$bi)\oplus(c$+$di) = (a+c)$ + $(c+d)i$
$(a$+$bi)\otimes(c$+$di) = (ac - bd)$ + $(ad+bc)i$
where $+$ and $-$ are the real number addition and subtraction, and + is merely a formal symbol.
Then you can show that you can identify the real number $a$ with the symbol $a$+$0i$; and that $(0$+$i)\otimes(0$+$i) = (-1)$+$0i$; etc. At that point you can start abusing notation and describing it as you do, using the same symbol for $+$, $\oplus$, and +. 
So... the method you propose (which was in fact how complex numbers were used at first) is just a bit more notationally abusive, while the method of ordered pairs is much more formal, giving a precise "substance" to complex numbers as "things" (assuming you think the plane is a "thing") and not just as "formal symbols". 
