One need only consult the history of algebra to find many informal
discussions on the existence and consistence of complex numbers.
Any *informal* attempt to justify the existence of $\mathbb C$
will face the same obstacles that existed in earlier times. Namely, the lack of any rigorous (set-theoretic) foundation makes it difficult to be precise - both syntactically and semantically. Nowadays the set-theoretic foundation of algebraic structures
is so subconscious that is is easy to overlook just how much power it provides
for such purposes. But this oversight is easily remedied. One need only consult
some of the older literature where even leading mathematicians struggled
immensely to rigorously define complex numbers. For example, see the
quote below by Cauchy and Hankel's scathing critique - which is guaranteed
to make your jaw drop! (Below is an excerpt from my post on the notion
of formal polynomial rings and their quotients).

A major accomplishment of the set-theoretical definition of
algebraic structures was to eliminate imprecise syntax and semantics.
Eliminating the syntactic polynomial term $\rm\ a+b\cdot x+c\cdot x^2\ $
and replacing it by its rigorous set-theoretic semantic reduction
$\rm\:(a,b,c,0,0,\ldots)\:$ eliminates many ambiguities. There can no longer be any doubt about the precise denotation of the symbols $\rm\: x,\; +,\;\cdot\:,$ or about the meaning of equality of polynomials, since, by set theoretic definition, tuples are equal iff their components are equal. The set-theoretic representation ("implementation") of these algebraic objects gives them *rigorous* meaning, reducing their semantics to that of set-theory.

Similarly for complex numbers $\rm\,a + b\cdot {\it i}\ $
and their set-theoretic representation by Hamilton as pairs of reals $\rm\,(a,b).\,$ Before Hamilton gave this semantic reduction of $\,\mathbb C\,$ to $\Bbb R^2,\,$ prior syntactic constructions (e.g. by Cauchy) as
formal expressions or terms $\rm\:a+b\cdot {\it i}\:$ were subject to heavy criticism regarding
the precise denotation of their constituent symbols, e.g.
precisely what is the meaning of the symbols $\rm\;{\it i},\, +,\, =\,?\, $
In modern language, Cauchy's construction of $\mathbb C$ is simply the
the quotient ring $\rm\:\mathbb R[x]/(x^2+1)\cong \Bbb R[{\it i}],\,$ which he described essentially
as real polynomial expressions modulo $\rm\:x^2+1\:$. However, in Cauchy's time
mathematics lacked the necessary (set-theory) foundations to
rigorously define the syntactic expressions comprising the
polynomial ring term-algebra $\rm\mathbb R[x]$, and its quotient ring of
congruence classes $\rm\:(mod\ x^2+1).\,$ The best that Cauchy could
do was to attempt to describe the constructions in terms of
imprecise *natural* (human) language, e.g, in 1821 Cauchy wrote:

In analysis, we call a symbolic expression any combination of
symbols or algebraic signs which means nothing by itself but
which one attributes a value different from the one it should
naturally be [...] Similarly, we call symbolic equations those
that, taken literally and interpreted according to conventions
generally established, are inaccurate or have no meaning, but
from which can be deduced accurate results, by changing and
altering, according to fixed rules, the equations or symbols
within [...] Among the symbolic expressions and equations
whose theory is of considerable importance in analysis, one
distinguishes especially those that have been called imaginary. $\quad$ -- Cauchy, Cours d'analyse,1821, S.7.1

While nowadays, using set theory, we can rigorously interpret such "symbolic expressions"
as terms of formal languages or term algebras, it was far too
imprecise in Cauchy's time to have any hope of making sense
to his colleagues, e.g. Hankel replied scathingly:

If one were to give a critique of this reasoning, we can not
actually see where to start. There must be something "which
means nothing," or "which is assigned a different value than
it should naturally be" something that has "no sense" or is
"incorrect", coupled with another similar kind, producing
something real. There must be "algebraic signs" - are these
signs for quantities or what? as a sign must designate something
- combined with each other in a way that has "a meaning." I do
not think I'm exaggerating in calling this an unintelligible
play on words, ill-becoming of mathematics, which is proud
and rightly proud of the clarity and evidence of its concepts. $\quad$-- Hankel

Thus it comes as no surprise that Hamilton's elimination
of such "meaningless" symbols - in favor of pairs of reals -
served as a major step forward in placing complex numbers on a
foundation more amenable to his contemporaries.
Although there was not yet any theory of sets in which to
rigorously axiomatize the notion of pairs, they were far easier
to accept naively - esp. given the already known closely
associated geometric interpretation of complex numbers.
Hamilton introduced pairs as 'couples' in 1837 [1]:

p. 6: The author acknowledges with pleasure that he agrees with
M. Cauchy, in considering every (so-called) Imaginary Equation
as a symbolic representation of two separate Real Equations:
but he differs from that excellent mathematician in his method
generally, and especially in not introducing the sign sqrt(-1)
until he has provided for it, by his Theory of Couples,
a possible and real meaning, as a symbol of the couple (0,1)

p. 111: But because Mr. Graves employed, in his reasoning, the
usual principles respecting about Imaginary Quantities, and
was content to prove the symbolical necessity without showing
the interpretation, or inner meaning, of his formulae, the
present Theory of Couples is published to make manifest that
hidden meaning: and to show, by this remarkable instance, that
expressions which seem according to common views to be merely
symbolical, and quite incapable of being interpreted, may pass
into the world of thoughts, and acquire reality and significance,
if Algebra be viewed as not a mere Art or Language, but as the
Science of Pure Time. $\quad$ -- Hamilton, 1837

Not until the much later development of set-theory was it explicitly realized
that ordered pairs and, more generally, n-tuples, serve a *fundamental* foundational role, providing the raw materials necessary to construct composite (sum/product) structures - the raw materials required for the above constructions of polynomial rings and their quotients.
Indeed, as Akihiro Kanamori wrote on p. 289 (17) of
his very interesting paper [2] on the history of set theory:

In 1897 Peano explicitly formulated the ordered pair using
$\rm\:(x, y)\:$ and moreover raised the two main points about the
ordered pair: First, equation 18 of his Definitions stated
the instrumental property which is all that is required of
the ordered pair:

$$\rm (x,y) = (a,b) \ \ \iff \ \ x = a \ \ and\ \ y = b $$

Second, he broached the possibility of reducibility, writing:
"The idea of a pair is fundamental, i.e., we do not know how
to express it using the preceding symbols."

Once set-theory was fully developed one had the raw materials
(syntax and semantics) to provide rigorous constructions of
algebraic structures and precise languages for term algebras. The polynomial ring $\rm\:R[x]\:$ is nowadays just a special case of much more general constructions of free algebras. Such equationally axiomatized algebras and their genesis via so-called 'universal mapping properties' are topics discussed at length in any course on Universal Algebra -
e.g. see Bergman [3] for a particularly lucid presentation.

[1] William Rowan Hamilton. Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time

Trans. Royal Irish Academy, v.17, part 1 (1837), pp. 293-422.)

http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/PureTime/PureTime.pdf

[2] Akihiro Kanamori. The Empty Set, the Singleton, and the Ordered Pair

The Bulletin of Symbolic Logic, Vol. 9, No. 3. (Sep., 2003), pp. 273-298.

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.95.9839

PS http://www.math.ucla.edu/~asl/bsl/0903/0903-001.ps

PDF http://ifile.it/b20c48j

[3] George M. Bergman. An Invitation to General Algebra and Universal Constructions.

PS http://math.berkeley.edu/~gbergman/245/

PDF http://ifile.it/yquj5w1