Mandelbrot set, but with power from 1 to ∞

This animation shows general Mandelbrot sets for which the exponent in the recurrence relation has discrete values in the interval [1, ∞). Generalizations of the Mandelbrot set obtained by changing the exponent are known as Multibrot sets. To compute the exponentiation of a complex number z ≠ 0 to a rational power p, I made the assumption that: zᵖ = exp(p * Log z), where Log z is the principal value of the complex logarithm of z. For example, (-1) ^ ½ has been evaluated to i. Last but not least, I also invited the unit disk to the party. Although that barely visible final frame was supposed to be rather a joke, that is actually how any extension of the Multibrot set to an exponent of ∞ would be represented if we worked with sequences in a metric space (C ∪ {∞}, d), where d can be any metric that preserves the regular limits of all sequences of complex numbers (such as the chordal or great circle distance on the Riemann sphere). In this case, since d(∞, ∞) = 0 by definition, if (zₙ)ₙ is a sequence such that zₙ = ∞ ∀n ≥ k ∈ N, then zₙ → ∞. Thus, by replacing zₙᵖ in the recurrence with lim as p → ∞ of zₙᵖ (denoted zₙ ^ ∞, where zₙ ∈ C ∪ {∞}), it turns out that, for a number c ∈ C: 1) if |c| is less than 1, then zₙ → c so we can naturally consider c a part of the Multibrot set; 2) if |c| is greater than 1 or c = 1, then zₙ → ∞ so we can naturally consider c a part of the complement of the Multibrot set; 3) if |c| = 1 and c ≠ 1, then c ^ ∞ does not exist and the membership of c...