Is it a good idea to adjust stock market indices to inflation? Well, why not. However, there's a glitch. The most usual inflation measure -- the Consumer Price Index -- is an awkward tool for this purpose. Asset prices should not be conflated with consumer prices. Equities are assets, consumer goods are not.
Discounting asset prices with the CPI is methodologically wrong. Do not expect any meaningful results if you combine the CPI inflation with equity prices, corporate earnings et cetera.
Is there a better approach? Yes. Using monetary inflation -- or money supply growth -- is the correct way to go. In other words, the question is "how would the stock market index look like if money supply remained constant?" (It's money supply, not consumer prices what matters to investors.)
The answer to the above question is as follows -- and it's a surprising chart indeed!
The takeaway of this exercise is, among others: