This corresponds to the familiar view of C as the `complex plane', a 2-dimensional vector space over R with basis 1, i.

This is an infinite-dimensional vector space. For instance, the numbers p,   for p prime numbers in N, form an infinite set of linearly independent elements. But not a basis, as elements such as e and are still not in their linear span.

This is the situation described before; we are dealing here with a 2-dimensional vector space over Z/2Z, and so with a field of 4 elements.