Generally, PowerShell's tail -f equivalent is Get-Content -Wait.
However, your clever combination of a Bash subshell ((...)) with a background process (&) has no direct PowerShell equivalent.
Instead, you must employ a loop to monitor the background process in PowerShell:
# Start the Docker command as a background job.
$jb = Start-Job { docker-compose -f docker-compose.yml logs -f }
# Loop until the data of interest is found.
while ($jb.HasMoreData) {
# Receive new data output by the background command, if any,
# and break out of the loop once the string of interest is found.
Receive-Job $jb -OutVariable output |
ForEach-Object { if ($_ -match "Initialization Complete") { break } }
# With a stream that is kept open, $jb.HasMoreData keeps reporting $true.
# To avoid a tight loop, we sleep a little whenever nothing was received.
if ($null -eq $output) { Start-Sleep -Seconds 1 }
}
# Clean up the background job, if it has completed.
if ($jb.Status -eq 'Complete') { Remove-Job $jb }