What is the dimension of $M_m$$_x$$_n$$\mathbb{C}$ when considered as a vector space over $\mathbb{R}$?
My approach: If I take an mxn matrix of complex entries and I want to write this as a linear combination of reals, I think that the dimension would have to be $2mn$ since the matrix could be broken down into two matrices, one containing the real entries and another containing the complex entries. E.g. if $a+bi$ is a component of the original mxn matrix, the $a$ part would be in one of the broken down matrices and the $bi$ part would be in the other. Then each of these two matrices can be broken down using $mn$ $mxn$ matrices which is how I get this conclusion. Is there anything I am doing wrong here I am very unsure for some reason?
Yes, the reasoning is correct. It would be 2mn (mn for the real part for each position and mn for the imaginary part for each position).