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Is there anything similar to Microsoft Powershell (an object-oriented shell built on the .NET framework) for Linux (possibly built on Java, GObject, or its own object type/nothing)?
edit: especially if similar to bash or powershell or cmd etc. syntax (=''standard'' shell syntax)
You should rethink why it is you think you need an object-oriented shell. That said, if you're set trying weird shells you can't go wrong with zoid. Unlike many of the other suggestions I see here it really is a shell. On the other hand, if you don't know or don't like Perl you probably won't be happy.
Python. No joking.
Scripting languages are scripting languages, and Python is a particularly nice one that many people find very approachable.
jq is not quite an object-oriented shell, but it provides some of the benefits which object-oriented shells may have; I use it a lot, together with shell scripts, for such tasks.
Hotwire
Even though this question is pretty old, I think its worth mentioning that in August 2016 Microsoft made Powershell open-source and cross platform. Instructions for installation are on github.
https://github.com/PowerShell/PowerShell
Perl, Python, and Ruby
Ok, I'm sure you already know that, but someone had to say it.
Perl is the oldest and most popular.
If you like objects, you will probably love Ruby. It has an elaborate object system inspired by Smalltalk.
Python has this cool block-structure-by-indent syntax.
Unix is a gold mine of advanced scripting tools...