As @Bastien_Siebman says, it’s your choice as to whether you keep your personal projects/tasks in your current organization as private items, or create a separate workspace.
(FYI it’s known as a workspace, not workplace.)
Separate orgs/workspaces are their own entities, not tied to one another, so the answer is “yes” but it’s not even a matter of moving anything over - you can just log into your personal workspace assuming you leave the organization you’re in.
Both the iOS and Android mobile apps let you switch between orgs/workspaces. On iOS you use the “Accounts” button at the bottom of the screen; on Android you use the avatar circle at the top right of the screen.
A few additional notes:
Asana’s My Tasks view is org/workspace specific, meaning it does not show tasks across multiple orgs/workspaces. So if you use a separate personal workspace, you won’t be able to see your work and personal tasks on the same list. If that’s important to you, that’s an argument in favor of putting your personal stuff as private items in your organization.
On the other hand, if you do that and then leave your organization, you’ll have to transfer all of your personal projects out of the organization into a personal workspace before you leave. Asana does not provide a way to do that transfer, BUT there is a free 3rd party tool that will do it, Ditto.