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".