The $20^{\text{th}}$ number will be the value of $20_{10}$ in base $4$. Thus, we see
\[20_{10} = (1) \cdot 4^2 + (1) \cdot 4^1 + 0 \cdot 4^0\]
\[= 110_{4}\]
$\boxed{E}$
~JustinLee2017
In our number system the base is ten. If the base were changed to four you would count as follows: 
$1,2,3,10,11,12,13,20,21,22,23,30,\ldots$ The twentieth number would be: 
$\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 44 \qquad\textbf{(D)}\ 104 \qquad\textbf{(E)}\ 110$