Experimental .NET Core Debugging in VS Code 

Learn to use Visual Studio, Visual Studio Team Services, Application Insights and Team Foundation Server to decrease rework, increase transparency into your application and increase the rate at which you can ship high quality software throughout the application lifecycle

情報源: Experimental .NET Core Debugging in VS Code | Microsoft Application Lifecycle Management

Visual Studio CodeでとうとうC#のデバッグが可能になったようです。