Math Help - A question on vector calculus

1. A question on vector calculus

Today while I was having my lecture on vector calculus, I suddenly thought of a vector of complex coefficients, something like this
$\displaystyle (a+bi)\text{\textbf{i}}+(c+di)\text{\textbf{j}}$.
Does this kind of stuff actually exist? If no, just ignore this naive question. If yes, your elaboration will be greatly appreciated. Thanks!

2. Originally Posted by Hinatico
Today while I was having my lecture on vector calculus, I suddenly thought of a vector of complex coefficients, something like this
$\displaystyle (a+bi)\text{\textbf{i}}+(c+di)\text{\textbf{j}}$.
Does this kind of stuff actually exist? If no, just ignore this naive question. If yes, your elaboration will be greatly appreciated. Thanks!
What is the complex plane?