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?

Right. For some flavor, the complex numbers are  perfectly represented by  matrices of this pattern:
$$  
 \left(  \begin{array}{rr}
  a  &  b  \\
   -b   &  a  
\end{array} 
  \right)  ,
  $$
with $a,b \in \mathbb R.$
In particular,
$$  1 \rightarrow
 \left(  \begin{array}{rr}
  1  &  0  \\
   0   &  1  
\end{array} 
  \right)  ,
 \; \; \; \; i  \rightarrow
 \left(  \begin{array}{rr}
  0  &  1  \\
   -1   &  0  
\end{array} 
  \right)  .
  $$
Once you accept the complex numbers, 
the quaternions are  perfectly represented by  matrices of this related pattern:
$$  
 \left(  \begin{array}{rr}
  \alpha  &  \beta  \\
   -\bar{\beta}   &  \bar{\alpha}  
\end{array} 
  \right)  ,
  $$
with $\alpha, \beta \in \mathbb C.$ In particular,
$$  1  \rightarrow
 \left(  \begin{array}{rr}
  1  &  0  \\
   0   &  1  
\end{array} 
  \right)  ,
 \; \; \; \; i  \rightarrow
 \left(  \begin{array}{rr}
  i  &  0  \\
   0   &  -i  
\end{array} 
  \right)  ,
 \; \; \; \; j  \rightarrow
 \left(  \begin{array}{rr}
  0  &  1  \\
   -1   &  0  
\end{array} 
  \right)  ,
 \; \; \; \; k  \rightarrow
 \left(  \begin{array}{rr}
  0  &  i  \\
   i   &  0  
\end{array} 
  \right)  .
  $$
If you prefer, you can replace the 2 by 2 complex matrices with 4 by 4 real matrices, whereupon all the entries in the matrices for $1,i,j,k$ are $0,\pm 1.$ Just a bit harder to remember. 
