I am new to this area. I have idea about complex and real numbers. Any help would be appreciated to find the dimension.
Firstly, you should specify what is your definition of dimension of sets.
Naturally, dimension stands for dimension of vector spaces i.e. the cardinality of a basis. One thing that is relevant to your question is that if you consider set of real numbers and set of complex numbers as vector spaces (with some operations, not merely sets!) then it can be proven easily that the vector space of complex numbers has dimension 2 over the vector space of real numbers. [Hint : Consider the polynomial $x^2 + 1 = 0 $ over the set of real numbers and then look at its roots. ]