[FEATURE] Always show the mascot, animate to imply activity of various kinds

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Keep the mascot around after project start
Use it to visually convey useful information such as current activity (web search, reading, writing, LLMing, it has a tip the user might find useful (soft onboarding).
Cheers!

Proposed Solution

Always show me the mascot, perhaps in the left hand column

Alternative Solutions

Maybe in the title bar?

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

Claude is doing something
I want to know what since I don't know whether to stop it as it's taking a long time and I fear the worst.

Additional Context

None