The phrase "between us" is the correct idiomatic expression meaning that the information should remain only with the two people involved.
Option (a) "secret" is redundant and grammatically awkward.
Option (b) "in myself" is not idiomatic and would not be used in standard English for keeping confidential information.
Option (c) "amongst us" implies sharing the secret among more than two people, which changes the meaning.
"Between us" specifically implies confidentiality limited to the speaker and listener, making it the most accurate choice.