Real vs Complex Dimension are different? Okay if $C$ is a complex vector space, (say with n dimensions) then if we consider $C_\mathbb{R}$ to be the same vector space except all scalar combinations are real valued vs complex. Why does a basis for $C_\mathbb{R}$ have 2n dimensions? 
It’s still the same space with a restriction on the scalar multiples it is closed under. 
 A: It's "the same" space in some sense, but when you have fewer scalars to pick from, it is harder to make a linear combination, and therefore harder for a set of vectors to be linearly dependent. 
Conversely, this makes it easier for a set of vectors to be independent, so it should not be surprising that you can have larger independent sets when you use a smaller scalar field.
A: The natural map $\Bbb R^2 \to\Bbb C, \ (x, y) \mapsto x+iy$ is an $\Bbb R$-linear isomorphism, hence $\Bbb R^2\cong\Bbb C$ as $\Bbb R$ vector spaces, and $\dim_{\Bbb R}\Bbb C=2$, and
$$\Bbb C^n\cong(\Bbb R^2)^n\cong\Bbb R^{2n}$$
over $\Bbb R$.
From the other view point, we cannot introduce multiplication with any complex number on an odd dimensional real vector space; in particular the natural multiplication by a complex number leads out of the one dimensional $\Bbb R$. 
A: The dimension of a vector space is basically the minimal number of basis vectors you need in order to span any vector you want, using vector addition and scalar multiplication.
To make things simple, let's look at $\Bbb C^3$ (although the same reasoning applies to any dimension).
If your space is over $\Bbb C$, what are your basis vectors? The usual choice would be $(1, 0, 0) , (0, 1, 0) , (0, 0, 1)$. Using scalar multiplication and vector addition, you can use these 3 vectors to span any vector in $\Bbb C^3$; If you want to express $(a+bi, c+di, e+fi)$ you can just use the linear combination $(a+bi)\cdot (1,0,0)+(c+di)\cdot (0,1,0)+(e+fi)\cdot (0,0,1)$.
But if you're over $\Bbb R$ instead of $\Bbb C$, you can't do that - you can only multiplies by real scalars. This means 3 basis vectors are just not enough - you need the basis $(1, 0, 0) , (0, 1, 0) , (0, 0, 1), (i,0,0), (0,i,0), (0,0,i)$ in order to span any vector using linear combinations of the basis vectors.
It's easy to understand if you think of the complex plane - you can't describe a number using only 1 coordiate - you need 2. The same goes for $\Bbb C^n$ for any $n$ - you need twice the information.
