Yes, the alternative is the Queens Indian, when White plays Nf3 instead of Nc3. Karpov specialized in Queens Indian if your looking for representative games.
What openings as black to use if white avoids Nimzo-Indian early?
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Alternately, if white plays Nf3 instead of Nc3, you can play the Bogo-Indian, which is what I prefer. It has a lot of similar themes to the Nimzo-Indian.
You can transpose to the QGD or semi-slav as well.
I'm currently learning the (complex) Nimzo-Indian and Bogo-Indian as black's response to white's 1.d4.
What are the main opening responses to be aware of if white intends to really deviate early from either of these two openings? I know a lot of lines transpose, so if there is a whole different system that black needs to know if playing Nimzo or Bogo, would be appreciated.