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.
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.
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.