Re: More complex numbers than reals?

Chris M. Thomasson explained on 7/9/2024 :

Are there "more" complex numbers than reals? It seems so, every real has its y, or imaginary, component set to zero. Therefore for each real there is an infinity of infinite embedding's for it wrt any real with a non-zero y axis? Fair enough, or really dumb? A little stupid? What do you think?

In a sense there are 'more' since the reals are all on the x axis line whereas the 2D R x R space is filled with complex numbers. R is contained in C. In another sense they are the same size set, Q being basically R by R in the same sense as Q being Z by Z).
Are there any other sizes of sets between countable Q and uncountable R? How about between uncountable R and uncountable C?