The questions you've presented pertain to the dimensions of the complex number field $\mathbb{C}$ viewed over different fields. Here are the answers:

1. Dimension of $\mathbb{C}$ over $\mathbb{C}$: The dimension of the vector space $\mathbb{C}$ over itself, $\mathbb{C}$, is 1. This is because the set $\{1\}$ can serve as a basis for $\mathbb{C}$ over $\mathbb{C}$.

2. Dimension of $\mathbb{C}$ over $\mathbb{R}$: The dimension of the complex numbers $\mathbb{C}$ over the real numbers $\mathbb{R}$ is 2. A basis for this vector space can be given by $\{1, i\}$, where $i$ is the imaginary unit, satisfying $i^2 = -1$.

3. Dimension of $\mathbb{C}$ over $\mathbb{Q}$: This case is more intricate. $\mathbb{C}$ over the rational numbers $\mathbb{Q}$ has an infinite dimension. This stems from the fact that any complex number can be expressed as $a + bi$, where $a$ and $b$ are real numbers, and each of these real numbers requires an infinite basis of rationals to be expressed as a rational linear combination.

Would you like more detailed explanations on any of these answers or have other questions?

Here are some related questions you might find interesting:

1. Why is the set $\{1, i\}$ a basis for $\mathbb{C}$ over $\mathbb{R}$?
2. How can we prove that $\mathbb{C}$ over $\mathbb{Q}$ has an infinite dimension?
3. What is a basis for $\mathbb{C}$ over $\mathbb{Q}$, and why is it infinite?
4. How do the properties of $\mathbb{C}$ change when considered over different fields?
5. What are the implications of a vector space having infinite dimension?

Tip: Understanding the concept of field extensions and the degrees of such extensions can greatly help in understanding the dimensions of vector spaces formed over different fields.