Output from save is unreadable. Output from dump or dput is ASCII and is readable to people who understand the structure of R objects, but I'm guessing you wanted it more conventionally arranged.
> apply(dataset, 1, function(x) paste(x, sep=",", collapse=","))
[1] "1,a,HI,c("a", "b")"
[2] "2,b,DD,c(2, 3, 4)"
[3] "3,c,gg,c(44, 33, 11, 22)"
[4] "4,d,ff,c("chr", "ID", "i", "II")"
The backslashes do not appear in the text-file output:
writeLines(con="test.txt", apply(dataset, 1, function(x) paste(x, sep=",", collapse=",")))
#-------output-----
1,a,HI,c("a", "b")
2,b,DD,c(2, 3, 4)
3,c,gg,c(44, 33, 11, 22)
4,d,ff,c("chr", "ID", "i", "II")