I think I recall seeing this question in a Halmos book on linear algebra, either "Finite Dimensional Vector Spaces" or the "Linear Algebra Problem Book", but I don't remember which, and I don't have them on hand.

Here are some subsets which satisfy 3 out of 4 conditions:

Jack Poulson already mentioned upper triangular matrices, which only violate (iv).

The set of all Hermitian matrices only violates (iii).

The set of Hermitian matrices $P_r$, where all eigenvalues are greater than some positive real $r$ is closed under addition — but not positive scaling — and every matrix can be written as an element of $P_r+(−P_r)+iP_r + (−iP_r)$. This set is a strict subset of $P$, and any element of $P \setminus P_r$ is not contained in $tP_rt^{-1}$ for any invertible $t \in M_n(\mathbb{C})$ (consider diagonalization).

The set of non-diagonalizable matrices with real, non-negative eigenvalues satisfies everything but (i). For $M_2$ explictly, consider matrices of the form
$$
A =
\left[ {\begin{array}{cc}
r_1 & z \\\
c\bar{z} & r_2 \\\
\end{array} } \right]
$$
where $r_1$, $r_2$ are real, $r_1 + r_2 > 0$, $z \neq 0$, and $c = -\left(\frac{r_1 - r_2}{2|z|}\right)^2$. Then $A$ has one repeated eigenvalue, $\frac{r_1 + r_2}{2}$, and one linearly independent eigenvector $(z, \frac{r_2-r_1}{2})$. The set of all such matrices satisfies (ii), (iii), (iv), and is not conjugate to $P$ — since everything in $P$ is diagonalizable — but is not closed under addition.