What is the best way to remove noise or outliers from a time series that has sudden shifts of levels? Consider the below time series. I have marked a typical outlier (that needs to be removed) and a typical level shift (which is genuine data, not an outlier).

The above time series is easy to handle by simple "differencing" and dropping any two consecutive values if their sum is much smaller (say 1/10th) of the individual values before differencing.
The problem starts when the spike stays on for a few samples, like a deep-pit - as shown in below time series chart.

The naked eye can see the dip as an outlier, but writing an algorithm gets increasingly complex as it needs sliding window means, standard deviations, etc. And the sliding window algo invariably fails in specific cases when the window size is smaller than the width of the deep-pit, or when there is a permanent change of level.
Can someone suggest a statistical algorithm to get rid of such spikes? [These are actually sensor errors due to a known problem (the battery voltage gets disconnected).]
Update
After Eoin's solution, which is close to working the following situation still fails.
All data points on the line between the sudden level shift are also classified as outliers.
See the two charts below, and it is self-evident that the marked point is not an outlier.


 A: A useful property here is that the outlier observations are different from both the mean of window of 20 or so points that come before them, and different from the window of points that come after. The level shift will be different from the window before, or different from the window after, but not different from both simultaneously. This means you can use a rule along the lines of
(abs(x - previous_window) > threshold) & (abs(x - next_window) > threshold)

Of course, you will still need to tweak the window widths and thresholds...
Code example
library(tidyverse)
n = 1000
times = seq(1, n)
true_signal = cumsum(rnorm(n)) # Random walk
yhat = ifelse(times < 750, true_signal, true_signal + 100) # Add jumps
y = rnorm(n, yhat, .1) # Add noise
y[500:505] = y[500:505] - 100
df = data.frame(t = times, true_signal, y)

ggplot(df, aes(t, y)) + geom_path()



# Detect true outliers
window = 20
threshold = 20

df$lagged_mean = map_dbl(times, function(t){
  mask = (df$t > t - window) & (df$t <= t)
  mean(df$y[mask])
})

df$leading_mean = map_dbl(times, function(t){
  mask = (df$t >= t) & (df$t < t + window)
  mean(df$y[mask])
})

df = mutate(df,
            lagged_delta = y - lagged_mean,
            leading_delta = y - leading_mean,
            is_outlier = abs(lagged_delta) > threshold & abs(leading_delta) > threshold)

df %>%
  mutate(is_outlier = is_outlier * 100) %>% # For visibility
  pivot_longer(-t) %>%
  ggplot(aes(t, value, color = name)) +
  geom_path()


Created on 2022-10-18 by the reprex package (v2.0.1)
