If you replace it, you must bring it up to current codes which requires an anti-scald technology tub/shower valve. Most of those are single handle. Those valves look identical to what was in my mother's house, also built in the 50's. To update it, I bought a renovation plate which covered the existing holes, then the new valve body was installed, and it's trim mounted on top of the renovation plate (sometimes called remodel plate - most of the manufactures make one). The plates are available in various shapes from a clean oval to some with various profiles and various finishes.
I happened to use a Delta R10000 rough-in valve with a set of 1700-series trim. The nice thing about that rough-in valve is that Delta makes three different series of trim/valves that work in it giving you a choice in varying costs and features from: simple pressure balanced, single handle valve; pressure balance with a volume control; thermostatically controlled. Any of these will fit in their R10000 rough-in valve, and you could change it later, should you change your mind. Most of these valves do not contain an integral diverter, so you'd replace the tub spout with one built-in. I like the Delta with the pull-down diverter - it's a big ring around the end of the spout that you pull down to divert the water; it springs back to tub function when you turn the water off. Their mounting system is quite flexible, too, which makes it a nice spout to use. They also come in various shapes and finishes, in plastic bodied to chrome (but CA may have limited that choice because of the low-lead, and that is extending the entire country). Lifetime finish warranty, so you don't' have to worry about that.
The big-box stores tend to sell these things as a kit which may include a valve, rough-in, trim, spout, and showerhead. If you go to a plumbing supply house (or online), you can buy the pieces you want and probably get similar pricing and more variety.