Looks like Perilla.

Thank you daisyduckworth. With some more information from Wikipedia, I believe you're right:

Perilla is a genus of annual herb that is a member of the mint family, Lamiaceae. The most common species is Perilla frutescens var. japonica or shiso which is mainly grown in India and East Asia. There are both green-leafed and purple-leafed varieties which are generally recognized as separate species by botanists. The leaves resemble stinging nettle leaves, being slightly rounder in shape. It is also widely known as the Beefsteak plant. In North America, it is increasingly commonly called by its Japanese name, shiso, in addition to being generally referred to as perilla. Its essential oils provide for a strong taste whose intensity might be compared to that of mint or fennel. It is considered rich in minerals and vitamins, has anti-inflammatory properties and is thought to help preserve and sterilize other foods. In Nepal and parts of India, it is called silam. Its seeds are ground with chili and tomatoes to make a savory dip/side dish.

In North America one of the purple varieties is sometimes known as Purple mint, Chinese basil, or Wild coleus (although it is not a mint, basil or coleus).

I was going to weed it out, as much as I could, but I think I'll keep it. Besides, it's already in the "mint patch".

It will get quite tall, and will produce seed-heads. Remove them!! They spread far and wide in huge numbers. I had to get rid of mine because it (and the amaranth) attracted plague numbers of grasshoppers which decimated my entire garden. It took about 2 years for the garden to recover from that.

So just be aware. I didn't like the taste of it much, but I liked it in the garden for the interesting foliage colour.

Used as a sweet-spicy flavouring for oriental dishes such as stirfries, with raw fish and sliced cucumber, in vegetable dishes, rice and soups. Red perilla is used as a red or pink food colouring, for pickling fruit and vegetables, especially preserved ginger and pickled sour plums, and as a dried powder to be used as a side dish with rice, as an ingredient in cake mixes and as a flavouring in beverages, as a condiment with sushimi. The sprouts can be used as a garnish. Green perilla, or oba, is used as a vegetable, for wrapping rice cake, in salads and tempura and goes well with sweet potato. The seeds of perilla are used to make oil and to flavour foods, especially pickles. Seeds (called egoma) can be used on baked goods, like sesame seeds. The flowerheads are used as a condiment. The oil from the seeds resembles Linseed oil and is used as a food flavouring in confectionery and sauces.