When I first learned about representation of a complex number by a point in a $2D$ plane, I wondered: what if it's redundant? What if a line is sufficient?
Apparently, it's not, but I still wonder: what is the proof of this? When was the fact first realized? Or was it totally obvious from the start of development of the theory of complex numbers?
History
Complex numbers were introduced by Cardan (Girolamo Cardano) in his Ars Magna in 1545.
They were however described only purely algebraically as a means for solving polynomial equations of degrees 3 or 4.
The first geometric description of $\mathbb C$ as points of a plane was given in 1799 by Wessel, a Danish cartographer, and independently by Argand, a Swiss-French bookshop owner, in 1806.
Argand was also the first to prove the "fundamental theorem of algebra" according to which a real non constant polynomial has at least one complex root and he was the first to notice that the theorem also applies to polynomials with complex coefficients.
Facts
The field $\mathbb C$ has dimension 2 (i.e. is a plane) as a real vector space : this is an easy result.
It also has dimension 2 as a differential manifold, which is a bit harder to show.
It has dimension 2 as a topological space: this is really hard since already the very notion of topological dimension is quite sophisticated.
Conclusion
To sum up and answer one of your questions:
No it was definitely not "totally obvious from the start of development of the theory of complex numbers" that they have dimension two.