The author develops a high accuracy algorithm for computing the complex zeros of the modified Bessel function of the second kind $K\sb{\nu}(z)$ and its derivatives with respect to z in the case of real or complex values of $\nu$. He states that the implementation of the algorithm has proved its high efficiency. It is shown that all he first zeros of the function $K\sb{\nu +2p}(z)$, $p=0,1,...$, as $p\to \infty$, lie on a straight line parallel to the real axis, and the distance between two adjacent zeros tends to a constant. All the second, third,... zeros have the same asymptotic property. The author constructs a table of all the complex zeros with nine significant figures of the function $K\sb{\nu}(z)$ for $\nu =2(1)20$, and of the function $K'\sb{\nu}(z)$ for $1/2\le \nu \le 20$. To determine the distribution of zeros of the derivatives of higher order, the loci of the zeros were constructed.