This, of course, refers to animals, not to plants. So Tom Tatoes are probably not a sign of the Apocalypse. If our world is destroyed, it will probably be for better reasons.
Recently, grafted vegetables have come to the market for gardeners. (Apparently, this method has been used commercially for some time- according to this article, 95% of the watermelons in Turkey are grafted onto pumpkin rootstock.) The most common examples in home gardening are grafted tomatoes. Heirloom tomatoes taste very good, but they tend to have weak and delicate root systems, and so they often die early or bear few marketable fruits. Scions of heirloom tomatoes are grafted on to commercial tomato rootstock, which is vigorous and disease-resistant. This is 100% muttar. However, Thompson and Morgan has recently begun selling a graft of a tomato and a potato, resulting in a plant which produces both crops. This is a fine and efficient use of one's garden, and it looks like it would be a lot of fun, but this is definitely a prohibited graft, and it is most probably assur to plant them. The fruit they produce is muttar, but planting and maintaining the grafted plant is a problem.
The problem is the issur of Kilei Zera'im, kilayim of plants. The Torah prohibits the grafting of dissimilar trees and vegetables. While some prohibitions of Kilayim- intermixing- apply only in Eretz Yisrael, the grafting prohibition applies even outside of Eretz Yisrael. This might be derabanan, it might be muttar when done by the hand of a gentile, there are plenty of "might"s. But I believe that the practical halacha is that it is assur to plant them.
If you're interested in the methodology, see here and here.
(The cite for the Chazon Ish regarding various kinds of Citrus fruits is Kilayim 3:7. The Chazon Ish there expresses uncertainty as to whether various citruses are considered dissimilar for purposes of Kilayim There is, however, no doubt in the world that tomatoes and potatoes are kilayim. While both are members of the family Solanum, that is irrelevant to the halachic determination of Kilayim. The connection is no more significant than the fact that apples and roses are members of Rosaceae.)
Sources, and I apologize for the abysmally boring names:
An article about the Tom Tato
New Zealand's Potato Tom
A discussion of the issur of grafting dissimilar plants
The author of the cited discussion, in the last paragraph, leans towards being mattir when done by a gentile. I disagree.