orthogonal complex plane If we have the 2 orthogonal vectors, which can also define a Euclidean plane (x-y plane):
 A = i + 0*j + 0*k
    B = 0*i + j + 0*k
Where their cross product gives the vector: 
C = A X B = 0*i + 0*j + k
And vectors (A,C) can be used to define the x-z plane and (B,C) to define the y-z plane, with planes x-z and y-z being orthogonal.
Question 1
Can we say that these two imaginary-vectors are also orthogonal?:
Ai = √(-1)*i + 0*j + 0*k
Bi = 0*i + √-1*j + 0*k
With their cross product giving the real vector:
    Ci = (0*i + 0*j – k)
Question 2
And can we then say, the vectors (Ai ,Ci) can be used to define a complex plane (call it  ia-z plane) and (Bi, Ci) to define a complex plane (call it ib - z plane) and these 2 planes are also orthogonal?
I’m unsure of the consistency of in the logic of the math.  I tried using right-quaternion vectors as a model, but I don’t think it’s the same.
 A: The answer to both of these questions is yes. When we talk about a vector space, we have to specify the "field that the vector space is over." Intuitively, this means "what field are the elements of a vector coming from." So you can have a 3-dimensional vector space with real elements (by far the most common) in which the result is as you describe. But you can also have a 3-dimensional vector space over the complex numbers, the rational numbers, or even finite fields! In general, these are usually denoted $F^k$ where $F$ is the field and $k$ is the dimension. So you've probably seen $\mathbb{R}^3$ plenty, but $\mathbb{Q}^3,\mathbb{C}^3$, and $\mathbb{F_5^3}$ are all perfectly good fields as well. Notice that in the third case, the vector space only has 15 elements! When you are using these vector spaces you have to be careful to use the right inner product, which is explained in this Wikipedia page.
To take $\mathbb{F}_5^3$ as an example, the vectors $(1,2,3)$ and $(3, 3, 2)$. Although you can define many different inner products on some vector spaces, the "usual" inner product on $\mathbb{R}^d$ usually has an obvious manifestation in the $F^d$. Notably, the reason that those two vectors are orthogonal is that $$1\cdot 3 + 2\cdot 3 + 3 \cdot 2= 0$$ holds where these operations are being considered inside of $\mathbb{F_5}$ (if you're not familair with $\mathbb{F_5}$, it's the integers modulo $5$. We just use the notation $\mathbb{F_p}$ when we want to stress the field structure of the integers modulo $p$)
There are even more exotic vector spaces, such as $L^2(\mathbb{C})$ which is a vector space that has infinite dimension! Elements can be thought of as functions from $\mathbb{C}$ to $\mathbb{C}$ that satisfy a certain criterion. The usual inner product there is defined by $\langle f,g\rangle=\sqrt{\int f(x)\overline{g(x)}dx}$, and the criterion for being in $L^2$ is that $\langle f,f\rangle = \int f(x)\overline{f(x)}dx=\sqrt{\int|f(x)|^2dx} < \infty$ where $|\cdot|$ is the usual norm on $\mathbb{C}$.
Although you sometimes have to be careful with infinite dimensional vector spaces (and things like $\mathbb{R}^\infty$ have ways of being interperted that make them perfectly good infinte dimensional vector spaces) most structural properties such as the ones you're talking about hold (or almost hold) in general vector spaces. In more general settings, we define a things like planes in terms of the degree of freedom that they have. The reason that looking at the set orthogonal to a vector in $\mathbb{R}^3$ gives a plane is actually because it leaves two coordinates not fixed (to see this, apply a change of basis that makes the first vector in your basis the vector that's defining the set).
This is generally true: Fixing all but two coordinates in a vector space is the definition of a plane. So to specify a plane in $\mathbb{R}^7$ you actually need FIVE vectors, since $7-5=2$. It turns out that some things you're used to being "properties of a plane" are actually "properties of what you get when you fix only one coordinate" and some of them are actually "properties of what you get when you fix all but two coordinates." This gets exciting when you think about less standard ways to represent coordinates. For example, specifying (in $\mathbb{R}^3$) that a point falls on a sphere of a given radius preserves a lot of the properties of a plane. But if you think about polar coordinates, the reason for this should be obvious.
