Have 1 always accessible search that is global to all contacts/content/rooms (mine and public)

**Is your suggestion related to a problem? Please describe.**

I use Element for work and often my work involves someone asking me to do something. When I have done it and need to update them, or ask follow up questions, I find it hard to remember which room the request started in. I click around a few rooms, scrolling up and down, trying to find the conversation where someone asked me to do something. I'm successful maybe 50% of the time, it depends on how memorable or familiar room names are to me. Even if it is familiar to me and I know the room, if it's dedicated to a project that involves a lot of collaborating, it can be hard to find the relevant discussion.

It's currently a bit difficult for me to actually move work projects forwards on Element.

**Describe the solution you'd like**
1 easy-to-find search bar that lets me look for anything.
For example, now that the "search" in the room list has been put at the top of the list, I think it should be more useful. Not just a filter for things underneath it, but something that can search through all contacts, rooms, room attachments..everything... I guess. Whether it stays in the exact same position or looks the exact same way isn't super relevant to the suggestion - the suggestion is just having 1 simple way to search to help find relevant things that are currently quite easy to lose.

**Describe alternatives you've considered**

- I considered just DMing the individuals who ask for things but often, for work, things require more collaboration and group feedback.
- Using another product to send work for feedback.

**Additional context**
Examples of roughly similar UI:

- https://dribbble.com/shots/11478574-Quick-Search
- https://dribbble.com/shots/8064813-Search-results
- Slack screenshot: