Real plane is denoted by and is commonly referred to as Cartesian plane. When we talk about we mean that is a vector space over . But when you view as Cartesian plane, then it’s not technically a vector space but rather an affine space, on which a vector space acts by translations, i.e. there is no canonical choice of where the origin should go in the space, because it can be translated anywhere.On the other hand, complex plane is denoted by and is commonly referred to as Argand plane. But when we talk about , we mean that is a field (by exploiting the tuple structure of elements) since there is only way to explicitly define the field structure on the set and that’s how we view as a field (if you allow axiom of choice, there are more possibilities; see this Math.SE discussion).
So, when we want to bother about the vector space structure of we refer to Cartesian plane and when we want to bother about the field structure of we refer to Argand plane. An immediate consequence of the above difference in real and complex plane is seen when we study multivariable analysis and complex analysis, where we consider vector space structure and field structure, respectively (see this Math.SE discussion for details). Hence the definition of differentiation of a function defined on is a special case of definition of differentiation of a function defined on .