Yes, but only in some of those languages popular with physicists, electrical engineers, and anyone of the sort who would have a beer with them. Complex numbers are especially useful in electronics, optics, and quantum theory for describing waves and any periodic phenomena. Fourier transforms use complex numbers and are the key to working with wavefunctions, designing filters, signal integrity in digital electronics, radio astronomy, and on and on...
The languages "D" (from Digital Mars), Python, IDL, Matlab (and Octave), APL, J, C (since C99), and Fortran have complex numbers built in. There are probably more. By "built in" we mean that one can write mathematical expressions with the same ease as when using plain floating point values, declare arrays of that type, and not having to work through some API or set of classes and methods even if syntactic sugar hides it. The compiler can optimize in ways not possible with a library.
Python is especially good to EEs by using "j" as the unit imaginary instead of the usual "i" used in math and physics.
Most other languages allow one to use complex numbers in the form of libraries. These include C, C++, Java, Ruby, perl, at least, among languages scientists and engineers have some likelihood of using.
I don't know about functional languages such as Haskell, in what manner they support complex numbers.
Programming for business, web apps, smartphones, and games rarely has any use for complex numbers. When they do, it's probably for some sort of audio or image analysis - science and engineering stuff - as part of some app. Languages that cater only to these industries might have complex libraries, but more likely don't support them at all. OTOH, it wouldn't surprise me if some smartass in the past had created a complex numbers library for COBOL.