The author develops a high accuracy algorithm for computing the complex zeros of the modified Bessel function of the second kind

${K}_{\nu}\left(z\right)$ 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

${K}_{\nu +2p}\left(z\right)$,

$p=0,1,\xb7\xb7\xb7$, 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

${K}_{\nu}\left(z\right)$ for

$\nu =2\left(1\right)20$, and of the function

${K}_{\nu}^{\text{'}}\left(z\right)$ 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.