You can firstly change by difference != 0 and then use na.locf to replace NAs by last available value recursively.

minimum_new <- function(data) {
  answer <- rep(NA, length(data))
  difference <- c(0, diff(data, lag = 1, differences = 1)) / 2
  answer[1] <- data[1]
  answer[difference != 0] <- data[difference != 0] - difference[difference != 0]
  answer <- zoo::na.locf(answer, na.rm = FALSE)
  answer
}

This version is faster for me by at least 2 times.

> check <- function(values) all(sapply(values[-1], function(x) identical(values[[1]], x)))
> bench <- microbenchmark::microbenchmark(loop = Minimum(data), vectorised = minimum_new(data), check=check)
Unit: microseconds
       expr      min       lq     mean   median       uq      max neval cld
       loop 1401.959 1415.552 1665.816 1457.274 1586.407 4620.835   100   b
 vectorised  742.325  758.183 1111.202  796.507 1383.268 2587.940   100  a 

With check it's also checks the equality of output.

You can firstly change by difference != 0 and then use na.locf to replace NAs by last available value recursively.

minimum_new <- function(data) {
  answer <- rep(NA, length(data))
  difference <- c(0, diff(data, lag = 1, differences = 1)) / 2
  answer[1] <- data[1]
  answer[difference != 0] <- data[difference != 0] - difference[difference != 0]
  answer <- zoo::na.locf(answer, na.rm = FALSE)
  answer
}

This version is faster for me by at least 2 times.

> data <- sample(10, 10000, replace = TRUE)
> check <- function(values) all(sapply(values[-1], function(x) identical(values[[1]], x)))
> bench <- microbenchmark::microbenchmark(loop = Minimum(data), vectorised = minimum_new(data), check=check)
Unit: microseconds
       expr      min       lq     mean   median       uq      max neval cld
       loop 1401.959 1415.552 1665.816 1457.274 1586.407 4620.835   100   b
 vectorised  742.325  758.183 1111.202  796.507 1383.268 2587.940   100  a 

With check it's also checks the equality of output.