Compare it with Cornus canadensis, which is a low growing, spreading dogwood.

Disclosure: I'm going by your description - I don't know what it is from the picture.

Here is a link that might be useful: Cornus canadensis

Thanks for your help, but this is definitely a small tree/shrub. I can see where my description would be deceiving. Sorry that the photo wasn't better. It was very difficult to get a good shot.

I've tried for several years to figure out what it is, but I have been unsuccessful.

I'm even wondering if the leader died off and the shoots growing horizontally are just the plant's adjustment to the loss.

Shrubby dogwood species, many dogwoods often grow at least partly in a layered or tiered fashion.

The colored stems make me think it is a 'colored bark' variety of Dogwood, that does not have the flowers Cornus florida or Kousa exhibit. The red stems resemble 'Arctic Fire'.
Several clones that are weeping forms of Cornus florida exist and are offered as rare plants. I believe one example is 'Weaver's Weeping'.
To my knowledge, they are rather poor bloomers, compared to the most common forms, but I am unsure.
If yours grows enough to bloom, and blooms pink colored, or blooms very well, it may be a keeper, that a business nursery would want to patent, or purchase from you.

Maybe this? Depending on how it was pruned/trained. I know mine grows much more horizontally than vertically. It is not one of the red twigs that grows like a cluster of vertical branches, nor is it bright red, but looks more like the color of the OP plant.

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Here is a link that might be useful: red twig?

I'd just let it develop and see what you get later, it does not look abnormal to me.

Olychick, I think you solved my problem. That link could be my plant. When I was outside today, doing some clean up, I went to trim some branches and, as in that link, they were rooted.

I love this group. You are all so knowledgeable.

Thanks to everyone for your suggestions.

Oh good! You can read online about pruning red twig dogwoods to promote a more vibrant bark color (on new growth) for winter interest. I don't prune mine often and so I get blossoms on it eventually, then prune it heavily to get it back in shape, get better color, but no blossoms for a few years.

I liked this article:

Here is a link that might be useful: Pruning red twig dogwood

Thanks. This link made me re-think removing this plant. It may become a favorite instead.