Complex numbers were used long before Gauss. They appeared for the first time in 16th century when people found a formula for solving cubic equations. One problem with this formula is that even for simplest equations like $x^3-x=0$ which have 3 real solutions, square roots of negative numbers occur in the formula (they cancel in the end, when you do calculation correctly). So use of the formula requires calculations with complex numbers, and people started investigating the rules of such calculations. They were called various names, "imaginary" numbers, "impossible" numbers, all these terms reflecting peoples confusion with them lasting till the beginning of 19th century.
Gauss found how to represent them geometrically, but even here he was not the first.
Predecessors were Jean-Robert Argand and Caspar Wessel. Various authors combined these three names with words such as plane, diagram etc.
See Argand diagram in Wikipedia.
Eventually the modern terminology emerged in 19th century: "complex numbers", meaning that they consist of two parts, real and imaginary. "Imaginary number" is used sometimes
to denote a complex number which is not real, or more frequently a number whose real part is zero (a.k.a "pure imaginary").
Gauss also investigated numbers of the form $m+ni$ where $m,n$ are integers. These are still called "Gauss integers", they have applications in questions about ordinary integers (number theory). The reason is that some primes can be factored using Gauss integers, like $5=(1+2i)(1-2i)$