If you do
$ ksh -n 'if [[ 1 -eq 1 ]]; then echo hi; fi'
you get the message
ksh: warning: line 1: -eq within [[...]] obsolete, use ((...))
as you've seen. Now try this:
$ ksh -n 'if (( 1 -eq 1 )); then echo hi; fi'
ksh: 1 -eq 1 : arithmetic syntax error
$ ksh -n 'if (( 1 == 1 )); then echo hi; fi'
Remember that the first message is only a warning. You can continue to use that form. I doubt that it will be removed since it would break too many existing scripts. By the way, this is accepted without a warning:
$ ksh -n 'if [ 1 -eq 1 ]; then echo hi; fi'
One of the main reasons that double parentheses is preferred is that the comparison operators are the more familiar
==, etc., (at least when compared to other languages).
Double parentheses also work in Bash and zsh. A related form, arithmetic substition, works in all of them, plus it's specified by POSIX.
$ a=$((3 * 4))
Korn, Bash and Z can also do it this way:
$ (( a = 3 * 4 ))
Even though Dash, as an example of a POSIX shell, doesn't support double parentheses comparisons in the form
if (( ... )), you can still do them using arithmetic substitution, but the result is the opposite from what you'd expect (this is also true for the others).
$ echo $(( 1 == 1 )); (( 1 == 1 )); echo $? # in Dash, use true instead of the second statement
$ echo $(( 1 == 2 )); (( 1 == 2 )); echo $? # in Dash, use false instead of the second statement