Can you declare a object literal type that allows unknown properties in typescript?

Yes, you can. Try this:

interface IBaz {
    baz: number;
    [key: string]: any;
}

function foo(bar: IBaz) : number {
    return bar.baz;
}

foo({ baz: 1, other: 2 });

Well, i hate answering my own questions, but the other answers inspired a little thought... This works:

function foo<T extends { baz: number }>(bar: T): void {
    console.log(bar.baz);
}

foo({baz: 1, other: 2});

If the known fields are coming from a generic type the way to allow wildcards is with T & {[key: string]: unknown}, any fields that are known must fit with the type's constraints and other fields are allowed (and considered type unknown)

Here is a sample:

type WithWildcards<T> = T & { [key: string]: unknown };

function test(foo: WithWildcards<{baz: number}>) {}

test({ baz: 1 }); // works
test({ baz: 1, other: 4 }); // works
test({ baz: '', other: 4 }); // fails since baz isn't a number

Then if you have a generic type T you can allow wildcard fields with WithWildCards<T>