Is there such thing as an imaginary (imaginary number)? In other words... is there such a thing that is to imaginary numbers what imaginary numbers are to real numbers? And could this be expressed as a "complex" type number? If a complex number is in the form x + yi, I guess this would be in the form of x + yi + zj?
Does that exist as a concept?

No there is no such thing.  To understand this you have to understand the imaginary unit: $i$ solves the equation $x^2 = -1$--that is $i^2 = -1$.  Right away there are is a "problem" because $-i$ solves this equation just as well as $+i$: $\left(-i\right)^2 = (-1)^2i^2 = +1i^2 = -1$ and $\left(+i\right)^2 = (+1)^2i^2 = i^2 =  -1$ so which do we choose?  Do we choose $-i$ or do we choose $+i$.  The answer is we choose neither and instead define $i$--we define $i$ such that $i^2 = -1$--it doesn't matter if we really mean $-i$ or $+i$--we just choose one of those two (we "choose" $+i$ because why carry around the burden of the negative when it's equivalent).
So to say there is an analogue to the complex unit $i$ would mean that there is some value of $x, y \in \mathbb{C}$ such that the following equation cannot be solved with complex numbers such that $x^2 = y$ where $x,y \in \mathbb{C}$.  This is not possible, there is a value $x \in \mathbb{C}$ such that $x^2 = y$ for all values in $y\in\mathbb{C}$.  We can go further and say that there exists at least one value $x \in \mathbb{C}$ such that for all values of $y \in \mathbb{C}$, $x^n = y$ where $n$ is a natural number (and you could expand to $n$ being a positive real value or a non-zero complex value).
I'm not familiar with these Qauternion values but these sound like higher dimension analogues.  I am strictly talking about $1$D values.  The imaginary number solves an insolvable problem from $\mathbb{R}^1$.  Once we extend $\mathbb{R}^1$ to include $\mathbb{C}^1$ all such problems become solvable.
