A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line.
A common point of ambiguity concerns the dimensionality of a complex line: while it is referred to as a "line" due to its one-dimensional nature in the context of C, over R it has two dimensions and is topologically equivalent to a real plane rather than a real line.