Relationship of basis vectors of the complex plane I am working on learning more about the connection of complex numbers and rotations in the context of rational geometry. Thanks ahead of time for any corrections on my best assertions.
Let $B$ represent the basis $\{(1, 0), (0, 1)\}$ for $\mathbb{R}^2$ where both axes are real.
Let $\mathbb{C}$ represent the complex plane. (My best assertion is) the basis vectors of the complex plane are $\{(1, 0), (0, i)\}$.
While the basis vectors of $\mathbb{C}$ are by definition linearly independent, there exists a relationship between the axes of $\mathbb{C}$ that does not exist in $B$ (i.e., $(0, 1i) \times (a, b) = (-b, a)$). 
What do I call this relationship between axes that can be different based on the choice of basis? I understand linear independence but not this seeming dependence between axes. Is this dependence the definition of rotation? It does not affect the linear independence of the basis vectors, yet it seems there is some sort of relationship between the axes of $\mathbb{C}$ that is not present in $B$.
 A: If you are identifying $\mathbb{R}^2$ with $\mathbb{C}$ using the correspondence $(a, b) \mapsto a + bi$ then $(1, 0)$ is identified with $1$, and $(0, 1)$ with $i$. The vectors $(1, 0)$ and $(0, i)$ don't lie in $\mathbb{C}$ but $\mathbb{C}^2$.
Now we can ask whether $\{1, i\}$ is linearly independent in $\mathbb{C}$. The answer depends on how you are viewing $\mathbb{C}$. If you are considering $\mathbb{C}$ as a real vector space, then the answer is yes; if $a, b \in \mathbb{R}$ and $a(1) + b(i) = 0$ then $a = 0$ and $b = 0$. If instead we view $\mathbb{C}$ as a complex vector space, then $\{1, i\}$ is not linearly independent as there are non-trivial solutions to $a(1) + b(i) = 0$ where $a, b \in \mathbb{C}$, namely $a = i$ and $b = -1$.
The vectors $(1, 0)$ and $(0, 1)$ are linearly independent in $\mathbb{R}^2$, as are $1$ and $i$ in $\mathbb{C}$ when viewed as a real vector space. What you point out is that $1$ and $i$ are related by a rotation, but a rotation in the complex plane can be performed by multiplication by a complex number (in particular, multiplication by $e^{i\theta} = \cos\theta + i\sin\theta$ is a rotation by $\theta$). Therefore, what you have noticed is a geometric reason why $1$ and $i$ are linearly dependent in $\mathbb{C}$ when viewed as a complex vector space.
