Is there a problem in defining a complex number by z=x+iy?

Step 1 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 rule $(a+bi)\oplus (c+di)=(a+c)+(c+d)i$ $(a+bi)\otimes (c+di)=(ac-bd)+(ad+bc)i$ 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.

Step 1 There is a completely rigorous way to do the construction you allude to in the last paragraph, namely by means of quotient rings. Indeed, $\mathbb{C}\cong \frac{\mathbb{R}[X]}{X^2+1}$ This generalises, for example, we can construct a commutative ring with elements of the form $x+yϵ$ where ${ϵ}^2=0$ The ring so constructed is emphatically not a field, but it is sometimes useful for doing symbolic differentiation.

Just set $i=(0,1)$ and $x=(x,0)$ for any real x, and the notation $x+iy$ is just a shorthand for the ordered pairs notation. Of course you could also choose $i=(0,-1)$