The evaluation of the modified Bessel function of the third kind of purely imaginary order K_ia(x) is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, non-oscillatory integral representations, asymptotic expansions and a continued fraction method, depending on the ranges of x and a. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method and the non-oscillatory integral representation can be used to accurately compute the function К_iа(х) in the range 0 <= a <= 200, 0 <= x <= 100; using a similar scheme the derivative K'_ia(x) can also be computed within these ranges.